Methods used to estimate deer density
Camera detection
The density of deer at a given site can be estimated using camera-trap distance sampling (CTDS). CTDS is a modified form of distance-sampling that allows us to infer the probability a given individual will be detected within the survey area (area in front of the camera). This detection probability is a function of the distance of the individual from the camera, whereby individuals entering the camera field of view further from the camera have a lower detection probability up to a given truncation distance from the camera where detection probability is near-zero.
An underlying assumption about CTDS is that the probability a deer will be available for detection at any given point location within the camera field of view is equal. Under this assumption, for a point transect, we take into account the total area for each distance-bin area, which increases at further distance bins. However, in this study we implement a novel method that considers group size of the detected species in the availability calculations. For larger groups, CTDS should account for the availability of the closest individual rather than the availability of all individuals. When group size increases it is more likely the the closest individual from the group is closer to the camera. If we assume that the triggering of the camera is dependent upon the closest individual than we must adjust our estimated availability to account for variable group sizes. If we do not adjust for group size and only use the distance to the closest individual for our DS models then we will likely underestimate the detection rate as distances will be smaller than reality. Alternatively, if we record distances to multiple individuals in the same photo and take an average or model them independently we would likely overestimate detection probability because individuals at further distances are recorded because a closer individual has triggered the camera trap. For more information on how group size is likely to effect CTDS abundance estimates a simulation study has been written here: https://justincally.github.io/blog/posts/2022-11-10-ctds-for-groups/
In this study we investigate two possible detection functions that may explain how detection rates ‘fall off’ over the distances from the camera. Firstly a half-normal detection function given by:
\[p = exp(-y^2/2\sigma^2) \] Hereby \(p\) is the probability of detection, \(y\) is the distance from the camera (midpoint of the detection bin), and \(\sigma\) is the shape parameter. Alternatively, we compare the fit of this detection function to a hazard-rate function, where detection probability is given by:
\[p = 1 - exp(-(y/\sigma)^{-\theta})\] In these hazard-rate models, an additional parameter (\(\theta\)) is estimated, which is scale parameter. Given that detection rate may not necessarily be uniform across sites we model the magnitude of the shape parameter across sites (i) as:
\[log(\sigma_i) = \alpha + \beta_{det} \times HU_i\] Where \(HU_i\) is the amount of herbaceous understorey measured surrounding the camera. \(\beta_{det}\) is the coefficient for the effect of herbaceous understorey on detection and \(\alpha\) is the intercept.
Using these estimates of detection probability in different bins we can then estimate the average detection probability of an individual at a given camera station. This can then be used to account for imperfect detection when fitting observed counts from the camera trap to parameters estimating the true abundance at a site.
Modelling density
Modelling abundnace using camera trap data
The density at a site is modelled as a function of the observed counts, relative frequency of group sizes, distance-sampling detection probability, survey effort (area in front of camera x time/snapshot moments the camera is deployed for), proportion of time within a 24-hour cycle that deer were active (\(activity\)). This parameter was estimated using the activity R package and included in the model using an informative beta-distribution prior. For a given species the number of detection’s of groups (1 or more deer) during the deployment is estimated with:
\[C_{n,j} = NegBinomial2(exp(log(\lambda_{n}) + log(p_{n,j}) + log(activity) + log(\epsilon gs_{n,j}) + log(\bar p)), \phi_n) \times survey\ area_n\]
Where \(C_{n,j}\) is the observed count of group size (\(j\)) at a site (\(n\)). \(\lambda_{n}\) is the true density at a site. In this model \(X_n\) are the values of the fixed-effect covariates, \(\beta\) are the fixed effect coefficients and \(\epsilon bioregion_n\) is the bioregion random effect drawn from a standard normal distribution:
\[\lambda_{n} = X_n \times \beta + \epsilon bioregion_n\]
The \(\epsilon gs_{n,j}\) is the proportional variation in group size ar a site, where \(\sum \epsilon gs_{n,j \in [1,max(group\ size)]} = 1\). Assuming that the rate of group sizes can vary between sites, we model the variation in group size between sites with a group-size level intercept (\(\zeta_j\)) and site-group-size level random effect (\(\epsilon site_{n,j}\)):
\[\epsilon psi_{nj} = exp(\zeta_j + \epsilon site_{n,j}) \] The proportional group size at a given site and for a given group size is thus given by:
\[\epsilon gs_{nj} = \frac{\epsilon psi_{nj}}{sum(\epsilon psi_{n})}\]
We opt for a negative binomial model to account for over-dispersion in counts/abundance. The parameterization of the negative binomial model implemented here (NegBinomial2) contains a mean as expressed by the log-sum of the first five terms. It also contains a dispersion parameter (\(\phi_n\)), which is the over-dispersion at the site. \(\phi_n\) is modelled as a function of bioregion:
\[\phi_n = exp(\chi mu_{bioregion_n} + \chi sd_{bioregion_n} \times \chi raw_{bioregion_n})\]
Supplementing the camera trap data with transect-level data
Our surveys involved the deployment of cameras at a site for usually 6-8 weeks as well as 150m transect searches that were walked up and back by a single individual for two transects, and one transect which was walked once by two observers. This gives us an approximate survey area at each site of 150m x 2 x 3 = 900m. Deer sign was recorded on these transects with pellets, footprints, rubbings, and wallows recorded as either present or absent.
This data provides supplemental presence-absence data that can be integrated into our model to help inform likely densities at sites where cameras did not record observations but one or more deer signs were detected along the transects.
The foundation for this integration is a Royle-Nichols model (Royle and Nichols 2003) that models the frequency of detection as a mixture of detection probability and abundance. Given that we also collect absolute abundance estimates via the camera trap data, relative abundance measures from this output can be transformed to absolute abundance and thus provide an avenue to integrate camera and transect-based data into a single model estimating abundance. The model can be expressed as:
\[N_n \sim NegBinomial2(\lambda_n, \bar\phi)\] Where \(\bar\phi\) is the over-dispersion term of the Royle-Nichols component of the model. Conditional detection probability is expressed based on the probability of detection from a given visit/survey type (\(j\)) (e.g. camera, pellet transect, footprint transect), and the number of individuals available to be detected at the site (\(N_n\)):
\[p_{nj} = 1 - (1 - r_{nj}) ^ {N_n}\] The model for this detection probability is then:
\[y_{nj} \sim Bernoulli(p_{nj})\] This model thus allows for \(\lambda_n\) to be modelled via the camera counts and the results of detections along the transect.
Finally, the Royle-Nichols model also allows us to estimate the rates at which with camera trap did not detect individuals at a site when they were present. Specifically we can use the average probability of detection of at least one individual at the camera trap, given individuals have expected abundance lambda (\(\bar p\)).
Setup
Load in relevant packages for analysis, additionally, connect to the database. Camera trap and site data is stored on the database.
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library(cmdstanr)
library(dplyr)
library(sf)
library(bayesplot)
library(loo)
library(gt)
library(terra)
library(caret)
library(tidyterra)
library(tidyr)
library(VicmapR)
library(kableExtra)
library(brms)
library(Distance)
library(ggrepel)
options(brms.backend = "cmdstanr")
check_cmdstan_toolchain(fix = TRUE, quiet = TRUE)
register_knitr_engine(override = FALSE)
options(mc.cores=8)
#### Database connection ####
con <- weda::weda_connect(password = keyring::key_get(service = "ari-dev-weda-psql-01",
username = "psql_user"), username = "psql_user")Custom Functions
Additional functions used in the data preparation, modelling and analysis are available in the /functions directory.
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# Source internal functions
sapply(list.files("functions", full.names = T), source, verbose = F)Prepare Data
Wrangle and format data for the STAN models for the various species.
Scope of models
Outline which species should be modeled, and which projects to source data from.
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# Species to run model for.
deer_species_all <- c("Cervus unicolor", "Dama dama", "Cervus elaphus", "Axis porcinus")
species_names <- c("Sambar Deer", "Fallow Deer", "Red Deer", "Hog Deer")
# Projects to select.
project_short_name <- c("hog_deer_2023", "StatewideDeer")
# Buffer for data extraction.
spatial_buffer <- 1000
# Covariate Rasters
raster_files <- "/Volumes/DeerVic\ Photos/Processed_Rasters"
# raster_files <- "data/prediction_raster"
prediction_raster <- "data/prediction_raster/statewide_raster.tif"
# For the integrated model we place limits on the maximum density of deer to integrate over. In cases where there are no detections on the camera this is limited to 15 per km2. In cases where deer were detected on the camera this is expanded to 50. We believe this is sufficiently high. Very high values of these will be less efficient.
n_max_no_det <- 15
n_max_det <- 30Camera locations
Download the camera locations from the database, this table outlines the locations and the deployment history of the cameras.
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cams_curated <- tbl(con, dbplyr::in_schema("camtrap", "curated_camtrap_operation")) %>%
dplyr::filter(ProjectShortName %in% !!project_short_name) %>% # Only retrieve cam records for hog deer 2023
dplyr::collect() %>%
dplyr::arrange(SiteID) %>%
sf::st_as_sf(., coords = c("Longitude", "Latitude"), crs = 4283)
n_site <- nrow(cams_curated)Formulas for detection and abundance
Here we outline formulas to be used in the models. The formulas account for the various fixed-effect parameters.
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#### Model formulas ####
#### Transect Formula: Survey Only ####
transect_formula <- ~Survey
#### Abundance Formula Options ####
ab_formula_1 <- ~ scale(sqrt(BareSoil)) + scale(sqrt(Nitrogen)) + scale(log(PastureDistance)) + scale(BIO15) + scale(sqrt(ForestEdge)) +
scale(sqrt(MRVBF)) + scale(sqrt(SLOPE)) + scale(sqrt(NonPhotosyntheticVeg))
ab_formula_2 <- ~ scale(sqrt(BareSoil)) + scale(sqrt(Nitrogen)) + scale(log(PastureDistance)) + scale(BIO15) + scale(sqrt(ForestEdge)) + scale(sqrt(MRVBF))
ab_formula_3 <- ~ scale(sqrt(BareSoil)) + scale(sqrt(Nitrogen)) + scale(log(PastureDistance)) + scale(BIO15) + scale(sqrt(ForestEdge))
ab_formula_4 <- ~ 1
#### Detection Formula: Distance-sampling ####
det_formula <- ~ scale(HerbaceousUnderstoryCover) # average detection across all sites Create model data
Using the prepare_model_data() function we generate the data for the STAN model. This function will:
1. Download data from the database 2. Format that data to match the distance-sampling bins
3. Organised the counts into sites and group sizes
4. Generate model matrix of the various sub-models (distance-sampling, abundance and transect) 5. Generate data for the prediction process 6. Generate data for the random effect (bioregion)
7. Generate data for regional predictions (indexing based on DEECA regions)
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if(!file.exists("data/multispecies_data.rds")) {
crown_land <- readRDS("data/crown_land.rds")
multispecies_data <- prepare_model_data_multispecies(species = deer_species_all,
projects = project_short_name,
buffer = spatial_buffer,
detection_formula = det_formula,
abundance_formula = ab_formula_1,
transect_formula = transect_formula,
con = con,
raster_dir = raster_files,
prediction_raster = prediction_raster,
n_max_no_det = n_max_no_det,
n_max_det = n_max_det,
crown_land = crown_land,
evaltransects = TRUE,
filter_behaviour = TRUE)
multispecies_data$keyfun <- 1 # 0 = HN, 1 = HZ
multispecies_data$raw_data[is.na(multispecies_data$raw_data)] <- 0
multispecies_data$transects[is.na(multispecies_data$transects)] <- 0
evc_groups <- readRDS("data/evc_groupnames.rds")
multispecies_data <- c(multispecies_data, evc_groups)
saveRDS(multispecies_data, "data/multispecies_data.rds")
} else {
multispecies_data <- readRDS("data/multispecies_data.rds")
}Model Execution
MCMC settings
Below we list the MCMC setting for our model. We run models on eight parallel chains for 400 iterations each (200 warm-up and 200 sampling). These setting provide us with 1,600 posterior draws (8 x 200).
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# STAN settings
ni <- 200 # sampling iterations
nw <- 200 # warm-up iterations
nc <- 8 # number of chainsRead in Models
These are the models used in the analysis. The first is an integrated model that requires transect and camera data. The second is a count-only model that just requires the camera data.
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functions {
/* Half-normal function
* Args:
* sigma: sigma
* midpoints: midpoints
* Returns:
* detection probability
*/
vector halfnorm(real sigma, vector midpoints) {
int bins = rows(midpoints);
vector[bins] p_raw; // detection probability
p_raw = exp(-(midpoints)^2/(2*sigma^2));
return p_raw;
}
/* Hazard function
* Args:
* sigma: sigma
* theta: theta
* midpoints: midpoints
* Returns:
* detection probability
*/
vector hazard(real sigma, real theta, vector midpoints) {
int bins = rows(midpoints);
vector[bins] p_raw; // detection probability
p_raw = 1 - exp(-(midpoints/sigma)^(-theta));
return p_raw;
}
vector prob_dist(real sigma, real theta, int keyfun, vector midpoints){
int bins = rows(midpoints);
vector[bins] out; // detection probability
if(keyfun == 0){
out = halfnorm(sigma, midpoints);
} else if(keyfun == 1){
out = hazard(sigma, theta, midpoints);
}
return out;
}
}
data {
int<lower=0> N; // number of observations
int<lower=0> S; // number of species
real delta; // bin width
int<lower=1> n_site; // site
int<lower=1> n_distance_bins; // distance bins
int<lower=1> n_gs; // number of group sizes
vector[n_gs] gs; // group sizes
vector[n_distance_bins] midpts; // midpt of each interval
real<lower=1> max_distance; // truncation distance
int<lower=1> max_int_dist; // max distance as an integer
real<lower=0> theta_frac; // fraction of camera view
array[n_site] int effort; // effort
array[n_site, n_gs, S] int n_obs; //number of observations
array[n_site, n_distance_bins, n_gs] int y; // observations matrix
// summary of whether species is known to be present at each site
int<lower = 0, upper = 1> any_seen[S, n_site];
// number of surveys at each site
int<lower = 0> n_survey[n_site];
// availability prior information
real<lower=0> bshape; // availability shape
real<lower=0> bscale; // availability scale
// camera trap detection parameters
int<lower=0> det_ncb; // number of covariates for detection model
matrix[n_site, det_ncb] det_model_matrix; // detection model matrix
array[n_gs, n_distance_bins] real pa ; // averaged availability for multiple individuals
// Abundance/occupancy model matrix
int<lower = 1> m_psi;
matrix[n_site, m_psi] X_psi;
// negbinom scale
// real reciprocal_phi_scale;
//transect level information
int<lower=1> trans; // total number of transects across all sites for all methods
array[S, trans] int<lower = 0, upper = 1> y2; // transect based binomial detections
int<lower = 0, upper = trans> start_idx[n_site];
int<lower = 0, upper = trans> end_idx[n_site];
int<lower=1> trans_det_ncb; // number of covariates for transect detection model
matrix[trans, trans_det_ncb] trans_pred_matrix; // transect detection model matrix
int<lower=1> n_max[n_site, S]; // max for poisson RN
// Prediction data
int<lower=1> npc; // number of prediction grid cells
matrix[npc, m_psi] X_pred_psi; // pred matrix
vector[npc] prop_pred; //offset
// bioregion RE
int<lower=1> np_bioreg;
int<lower=1> site_bioreg[n_site];
int<lower=1> pred_bioreg[npc];
// region data
int<lower=1> np_reg;
int<lower=1> site_reg[n_site];
int<lower=1> pred_reg[npc];
// key function, 0 = halfnorm
int keyfun;
}
transformed data {
// vector[n_distance_bins] pi; // availability for point
vector[n_site] survey_area;
array[S, n_site] real cam_seen;
for(i in 1:n_site) {
survey_area[i] = theta_frac * effort[i] * pi() * (max_distance/1000)^2;
for(s in 1:S) {
cam_seen[s,i] = sum(n_obs[i,,s]);
}
}
}
parameters {
// abundance parameters
//array[n_site] simplex[n_gs] eps_ngs; // random variation in group size
array[S] vector[m_psi] beta_psi;
vector[det_ncb] beta_det;
real log_theta;
// transect detection parameters
vector[trans_det_ncb] beta_trans_det;
// temporal availability parameters
real<lower=0, upper=1> activ;
// bioregion RE
array[S] real<lower=0> bioregion_sd;
array[S] vector[np_bioreg] bioregion_raw;
// eps group size params
array[S] vector[n_gs] zeta;
array[S] matrix[n_site, n_gs] eps_raw;
array[S] real<lower=0> grp_sd;
// od
array[S] real od_mu;
real odRNmu;
}
transformed parameters {
// random effects
array[S] vector[np_bioreg] eps_bioregion; // bioregion random effect
// distance parameters
array[n_site] real log_sigma;
array[n_site] real sigma;
array[n_site] vector[n_distance_bins] p_raw; // detection probability
array[n_site, n_distance_bins, n_gs] real p_raw_scale; // detection probability for point independence model and accounting for availability
array[n_site, n_distance_bins, n_gs] real<upper=0> log_p_raw; // log detection probability accounting for availability
array[n_site, n_gs] real log_p; // log probability with summed exponential for the multinomial logit model
array[n_site, n_gs] real<lower=0,upper=1> p; // log probability with summed exponential for the multinomial logit model
// abundance parameters
array[S, n_site, n_gs] real<lower=0> lambda;
// activity parameters
real log_activ = log(activ);
vector[trans] logit_trans_p = trans_pred_matrix * beta_trans_det; // observation process model
// lp_site for RN model
array[S, n_site] real lp_site;
vector<lower=0,upper=1>[trans] r = inv_logit(logit_trans_p);
array[S] vector[n_site] log_lambda_psi;
// eps group size
array[S, n_site] simplex[n_gs] eps_ngs; // random variation in group size
array[S, n_site] vector[n_gs] epsi;
array[S] matrix[n_site, n_gs] eps_site;
real<lower=0> theta = exp(log_theta);
array[S] real od; // bioregion random effect
real odRN = exp(odRNmu); // bioregion random effect
array[S, n_site] real<lower=0, upper=1> pbar;
for(s in 1:S) {
for(b in 1:np_bioreg) {
eps_bioregion[s,b] = bioregion_sd[s] * bioregion_raw[s,b];
}
od[s] = exp(odRNmu + od_mu[s]);
}
for(n in 1:n_site) {
log_sigma[n] = det_model_matrix[n,] * beta_det;
sigma[n] = exp(log_sigma[n]);
p_raw[n] = prob_dist(sigma[n], theta, keyfun, midpts);
for (i in 1:n_distance_bins) {
// assuming a half-normal detection fn from line
// p_raw[n,i] = exp(-(midpts[i])^2/(2*sigma[n]^2)); // half-normal function (pg 419 of AHM)
// hazard function
// p_raw[n,i] = 1 - exp(-(midpts[i]/theta)^(-sigma[n])); // hazard function (pg 507 of AHM)
for(j in 1:n_gs) {
p_raw_scale[n,i,j] = p_raw[n,i]*pa[j,i]; // pr(animal occurs and is detected in bin i)
log_p_raw[n,i,j] = log(p_raw_scale[n,i,j]);
}
}
for(s in 1:S) {
vector[n_max[n,s]+1] Nlik;
vector[n_max[n,s]+1] gN;
vector[n_max[n,s]+1] pcond;
log_lambda_psi[s,n] = X_psi[n,] * beta_psi[s] + eps_bioregion[s,site_bioreg[n]];
for(j in 1:n_gs) {
eps_site[s, n,j] = grp_sd[s] * eps_raw[s,n,j];
epsi[s,n,j] = exp(zeta[s,j] + eps_site[s,n,j]);
}
eps_ngs[s,n,] = epsi[s,n,]/sum(epsi[s,n,]);
// p-bar
for(k in 1:(n_max[n,s]+1))
Nlik[k] = exp(neg_binomial_2_log_lpmf(k-1 | log_lambda_psi[s,n], odRN));
gN = Nlik/sum(Nlik);
for(k in 1:(n_max[n,s]+1))
pcond[k] = (1 - (1 - r[start_idx[n]])^(k-1)) * gN[k];
pbar[s,n] = sum(pcond);
}
for(j in 1:n_gs) {
log_p[n,j] = log_sum_exp(log_p_raw[n,,j]);
p[n,j] = exp(log_p[n,j]);
// model site abundance
for(s in 1:S) {
lambda[s,n,j] = exp(log_lambda_psi[s,n] + log_p[n,j] + log_activ + log(eps_ngs[s,n,j])) .* survey_area[n];
}
}
// Royle-Nichols implementation in STAN (looping over possible discrete values of N)
// https://discourse.mc-stan.org/t/royle-and-nichols/14150
// https://discourse.mc-stan.org/t/identifiability-across-levels-in-occupancy-model/5340/2
for(s in 1:S) {
if (n_survey[n] > 0) {
vector[n_max[n,s]] lp;
if(any_seen[s,n] == 0){ // not seen
lp[1] = log_sum_exp(neg_binomial_2_log_lpmf(0 | log_lambda_psi[s,n], odRN), neg_binomial_2_log_lpmf(1 | log_lambda_psi[s,n], odRN) + bernoulli_lpmf(y2[s,start_idx[n]:end_idx[n]] | r[start_idx[n]:end_idx[n]]));
} else {
lp[1] = neg_binomial_2_log_lpmf(1 | log_lambda_psi[s,n], odRN) + bernoulli_lpmf(y2[s,start_idx[n]:end_idx[n]] | r[start_idx[n]:end_idx[n]]);
}
for (k in 2:n_max[n,s]){
lp[k] = neg_binomial_2_log_lpmf(k | log_lambda_psi[s,n], odRN) + bernoulli_lpmf(y2[s,start_idx[n]:end_idx[n]] | 1-(1-r[start_idx[n]:end_idx[n]])^k);
}
lp_site[s,n] = log_sum_exp(lp);
} else {
lp_site[s, n] = 0;
}
}
}
}
model {
for(s in 1:S) {
beta_psi[s] ~ normal(0, 3); // prior for intercept in poisson model
bioregion_sd[s] ~ normal(0,2);
bioregion_raw[s,] ~ normal(0,1);
to_vector(eps_raw[s,,]) ~ std_normal();
grp_sd[s] ~ normal(0, 1);
zeta[s,] ~ normal(0, 2);
// od
od_mu[s] ~ normal(0,1);
}
odRNmu ~ normal(0,1);
beta_trans_det ~ normal(0, 2); // beta for transect detection
beta_det ~ normal(0, 4); // prior for sigma
activ ~ beta(bshape, bscale); //informative prior
log_theta ~ normal(0,2); // prior for theta
for(n in 1:n_site) {
for(j in 1:n_gs) {
for(s in 1:S) {
target += neg_binomial_2_lpmf(n_obs[n,j,s] | lambda[s,n,j], od[s]);
}
y[n,,j] ~ multinomial_logit(to_vector(log_p_raw[n,,j]));
}
for(s in 1:S) {
target += lp_site[s, n];
}
}
}
generated quantities {
array[S, n_site,n_gs] real n_obs_pred;
array[S, n_site, n_gs] real n_obs_true;
array[S, n_site] real N_site;
array[S, n_site] real N_site_pred;
array[n_site, max_int_dist+1] real DetCurve;
array[n_site, n_gs] real log_lik1;
array[S, n_site, n_gs] real log_lik2;
array[n_site, n_gs] real log_lik2_site;
array[S, n_site] real log_lik2_species;
array[n_site] real log_lik;
array[n_site] real log_lik_det;
//array[S, n_site] real Site_lambda;
real av_gs[S];
array[S] simplex[n_gs] eps_gs_ave;
array[S, npc] real pred;
//array[S, npc] real pred_trunc;
array[S, np_reg] real Nhat_reg;
array[S, np_bioreg] real Nhat_bioreg;
// array[np_reg] real Nhat_reg_design;
real Nhat[S];
//real Nhat_trunc[S];
real Nhat_sum;
int trunc_counter[S];
trunc_counter[S] = 0;
for(s in 1:S) {
eps_gs_ave[s] = exp(zeta[s])/sum(exp(zeta[s]));
av_gs[s] = sum(gs .* eps_gs_ave[s]); //average group size
}
for(n in 1:n_site) {
for(j in 1:n_gs) {
log_lik1[n,j] = multinomial_logit_lpmf(y[n,,j] | to_vector(log_p_raw[n,,j])); //for loo
for(s in 1:S) {
log_lik2[s,n,j] = neg_binomial_2_lpmf(n_obs[n,j,s] | lambda[s,n,j], od[s]); //for loo
n_obs_true[s, n, j] = gs[j] * (neg_binomial_2_log_rng(log_lambda_psi[s,n] + log(eps_ngs[s,n,j]), od[s]));
n_obs_pred[s, n,j] = gs[j] * neg_binomial_2_rng(lambda[s,n,j], od[s]);
}
log_lik2_site[n, j] = log_sum_exp(log_lik2[,n,j]);
}
// get loglik on a site level
log_lik_det[n] = log_sum_exp(log_lik1[n,]);
log_lik[n] = log_sum_exp(log_sum_exp(log_lik_det[n],
log_sum_exp(log_lik2_site[n,])), log_sum_exp(lp_site[,n]));
for(s in 1:S) {
//Site_lambda[s,n] = exp(log_lambda_psi[s,n]);
log_lik2_species[s, n] = log_sum_exp(log_sum_exp(log_lik2[s,n,]), lp_site[s,n]);
N_site[s,n] = sum(n_obs_true[s,n,]);
N_site_pred[s,n] = sum(n_obs_pred[s,n,]);
}
// loop over distance bins
if(keyfun == 0) {
for(j in 0:max_int_dist) { // get DS predictions for distance 0 to max bin distance
DetCurve[n, j+1] = exp(-(j+0.5)^2/(2*sigma[n]^2)); // half normal
}
} else if(keyfun == 1) {
for(j in 0:max_int_dist) { // get DS predictions for distance 0 to max bin distance
DetCurve[n, j+1] = 1 - exp(-((j+0.5)/sigma[n])^(-theta)); //hazard rate
}
}
}
for(s in 1:S) {
for(i in 1:np_reg) Nhat_reg[s,i] = 0;
for(i in 1:np_bioreg) Nhat_bioreg[s,i] = 0;
for(i in 1:npc) {
pred[s,i] = neg_binomial_2_log_rng(X_pred_psi[i,] * beta_psi[s] + eps_bioregion[s, pred_bioreg[i]], od[s]) * prop_pred[i] * av_gs[s]; //offset
if(pred[s,i] > max(N_site[s,])) {
trunc_counter[s] += 1;
}
// for Hog Deer limit to Gippsland region
if(pred_reg[i] != 2 && s == 4) {
pred[s,i] = 0;
}
Nhat_reg[s,pred_reg[i]] += pred[s,i];
Nhat_bioreg[s,pred_bioreg[i]] += pred[s,i];
}
Nhat[s] = sum(pred[s,]);
//Nhat_trunc[s] = sum(pred_trunc[s,]);
}
Nhat_sum = sum(Nhat);
}Show code
functions {
/* Half-normal function
* Args:
* sigma: sigma
* midpoints: midpoints
* Returns:
* detection probability
*/
vector halfnorm(real sigma, vector midpoints) {
int bins = rows(midpoints);
vector[bins] p_raw; // detection probability
p_raw = exp(-(midpoints)^2/(2*sigma^2));
return p_raw;
}
/* Hazard function
* Args:
* sigma: sigma
* theta: theta
* midpoints: midpoints
* Returns:
* detection probability
*/
vector hazard(real sigma, real theta, vector midpoints) {
int bins = rows(midpoints);
vector[bins] p_raw; // detection probability
p_raw = 1 - exp(-(midpoints/sigma)^(-theta));
return p_raw;
}
vector prob_dist(real sigma, real theta, int keyfun, vector midpoints){
int bins = rows(midpoints);
vector[bins] out; // detection probability
if(keyfun == 0){
out = halfnorm(sigma, midpoints);
} else if(keyfun == 1){
out = hazard(sigma, theta, midpoints);
}
return out;
}
}
data {
int<lower=0> N; // number of observations
int<lower=0> S; // number of species
real delta; // bin width
int<lower=1> n_site; // site
int<lower=1> n_distance_bins; // distance bins
int<lower=1> n_gs; // number of group sizes
vector[n_gs] gs; // group sizes
vector[n_distance_bins] midpts; // midpt of each interval
real<lower=1> max_distance; // truncation distance
int<lower=1> max_int_dist; // max distance as an integer
real<lower=0> theta_frac; // fraction of camera view
array[n_site] int effort; // effort
array[n_site, n_gs, S] int n_obs; //number of observations
array[n_site, n_distance_bins, n_gs] int y; // observations matrix
// summary of whether species is known to be present at each site
int<lower = 0, upper = 1> any_seen[S, n_site];
// number of surveys at each site
int<lower = 0> n_survey[n_site];
// availability prior information
real<lower=0> bshape; // availability shape
real<lower=0> bscale; // availability scale
// camera trap detection parameters
int<lower=0> det_ncb; // number of covariates for detection model
matrix[n_site, det_ncb] det_model_matrix; // detection model matrix
array[n_gs, n_distance_bins] real pa ; // averaged availability for multiple individuals
// Abundance/occupancy model matrix
int<lower = 1> m_psi;
matrix[n_site, m_psi] X_psi;
// negbinom scale
// real reciprocal_phi_scale;
//transect level information
int<lower=1> trans; // total number of transects across all sites for all methods
array[S, trans] int<lower = 0, upper = 1> y2; // transect based binomial detections
int<lower = 0, upper = trans> start_idx[n_site];
int<lower = 0, upper = trans> end_idx[n_site];
int<lower=1> trans_det_ncb; // number of covariates for transect detection model
matrix[trans, trans_det_ncb] trans_pred_matrix; // transect detection model matrix
int<lower=1> n_max[n_site, S]; // max for poisson RN
// Prediction data
int<lower=1> npc; // number of prediction grid cells
matrix[npc, m_psi] X_pred_psi; // pred matrix
vector[npc] prop_pred; //offset
// bioregion RE
int<lower=1> np_bioreg;
int<lower=1> site_bioreg[n_site];
int<lower=1> pred_bioreg[npc];
// region data
int<lower=1> np_reg;
int<lower=1> site_reg[n_site];
int<lower=1> pred_reg[npc];
// evc data
int<lower=1> np_evc;
int<lower=1> site_evc[n_site];
int<lower=1> pred_evc[npc];
// key function, 0 = halfnorm
int keyfun;
}
transformed data {
// vector[n_distance_bins] pi; // availability for point
vector[n_site] survey_area;
array[S, n_site] real cam_seen;
for(i in 1:n_site) {
survey_area[i] = theta_frac * effort[i] * pi() * (max_distance/1000)^2;
for(s in 1:S) {
cam_seen[s,i] = sum(n_obs[i,,s]);
}
}
}
parameters {
// abundance parameters
//array[n_site] simplex[n_gs] eps_ngs; // random variation in group size
array[S] vector[m_psi] beta_psi;
vector[det_ncb] beta_det;
real log_theta;
// transect detection parameters
vector[trans_det_ncb] beta_trans_det;
// temporal availability parameters
real<lower=0, upper=1> activ;
// bioregion RE
array[S] real<lower=0> bioregion_sd;
array[S] vector[np_bioreg] bioregion_raw;
// evc RE
real<lower=0> evc_sd;
array[S] vector[np_evc] evc_raw;
// eps group size params
array[S] vector[n_gs] zeta;
array[S] matrix[n_site, n_gs] eps_raw;
array[S] real<lower=0> grp_sd;
// od
array[S] real od_mu;
real odRNmu;
}
transformed parameters {
// random effects
array[S] vector[np_bioreg] eps_bioregion; // bioregion random effect
array[S] vector[np_evc] eps_evc; // bioregion random effect
// distance parameters
array[n_site] real log_sigma;
array[n_site] real sigma;
array[n_site] vector[n_distance_bins] p_raw; // detection probability
array[n_site, n_distance_bins, n_gs] real p_raw_scale; // detection probability for point independence model and accounting for availability
array[n_site, n_distance_bins, n_gs] real<upper=0> log_p_raw; // log detection probability accounting for availability
array[n_site, n_gs] real log_p; // log probability with summed exponential for the multinomial logit model
array[n_site, n_gs] real<lower=0,upper=1> p; // log probability with summed exponential for the multinomial logit model
// abundance parameters
array[S, n_site, n_gs] real<lower=0> lambda;
// activity parameters
real log_activ = log(activ);
vector[trans] logit_trans_p = trans_pred_matrix * beta_trans_det; // observation process model
// lp_site for RN model
array[S, n_site] real lp_site;
vector<lower=0,upper=1>[trans] r = inv_logit(logit_trans_p);
array[S] vector[n_site] log_lambda_psi;
// eps group size
array[S, n_site] simplex[n_gs] eps_ngs; // random variation in group size
array[S, n_site] vector[n_gs] epsi;
array[S] matrix[n_site, n_gs] eps_site;
real<lower=0> theta = exp(log_theta);
array[S] real od; // bioregion random effect
real odRN = exp(odRNmu); // bioregion random effect
array[S, n_site] real<lower=0, upper=1> pbar;
for(s in 1:S) {
for(b in 1:np_bioreg) {
eps_bioregion[s,b] = bioregion_sd[s] * bioregion_raw[s,b];
}
od[s] = exp(odRNmu + od_mu[s]);
for(k in 1:np_evc) {
eps_evc[s,k] = evc_sd * evc_raw[s,k];
}
}
for(n in 1:n_site) {
log_sigma[n] = det_model_matrix[n,] * beta_det;
sigma[n] = exp(log_sigma[n]);
p_raw[n] = prob_dist(sigma[n], theta, keyfun, midpts);
for (i in 1:n_distance_bins) {
// assuming a half-normal detection fn from line
// p_raw[n,i] = exp(-(midpts[i])^2/(2*sigma[n]^2)); // half-normal function (pg 419 of AHM)
// hazard function
// p_raw[n,i] = 1 - exp(-(midpts[i]/theta)^(-sigma[n])); // hazard function (pg 507 of AHM)
for(j in 1:n_gs) {
p_raw_scale[n,i,j] = p_raw[n,i]*pa[j,i]; // pr(animal occurs and is detected in bin i)
log_p_raw[n,i,j] = log(p_raw_scale[n,i,j]);
}
}
for(s in 1:S) {
vector[n_max[n,s]+1] Nlik;
vector[n_max[n,s]+1] gN;
vector[n_max[n,s]+1] pcond;
log_lambda_psi[s,n] = X_psi[n,] * beta_psi[s] + eps_bioregion[s,site_bioreg[n]] + eps_evc[s,site_evc[n]];
for(j in 1:n_gs) {
eps_site[s, n,j] = grp_sd[s] * eps_raw[s,n,j];
epsi[s,n,j] = exp(zeta[s,j] + eps_site[s,n,j]);
}
eps_ngs[s,n,] = epsi[s,n,]/sum(epsi[s,n,]);
// p-bar
for(k in 1:(n_max[n,s]+1))
Nlik[k] = exp(neg_binomial_2_log_lpmf(k-1 | log_lambda_psi[s,n], odRN));
gN = Nlik/sum(Nlik);
for(k in 1:(n_max[n,s]+1))
pcond[k] = (1 - (1 - r[start_idx[n]])^(k-1)) * gN[k];
pbar[s,n] = sum(pcond);
}
for(j in 1:n_gs) {
log_p[n,j] = log_sum_exp(log_p_raw[n,,j]);
p[n,j] = exp(log_p[n,j]);
// model site abundance
for(s in 1:S) {
lambda[s,n,j] = exp(log_lambda_psi[s,n] + log_p[n,j] + log_activ + log(eps_ngs[s,n,j])) .* survey_area[n];
}
}
// Royle-Nichols implementation in STAN (looping over possible discrete values of N)
// https://discourse.mc-stan.org/t/royle-and-nichols/14150
// https://discourse.mc-stan.org/t/identifiability-across-levels-in-occupancy-model/5340/2
for(s in 1:S) {
if (n_survey[n] > 0) {
vector[n_max[n,s]] lp;
if(any_seen[s,n] == 0){ // not seen
lp[1] = log_sum_exp(neg_binomial_2_log_lpmf(0 | log_lambda_psi[s,n], odRN), neg_binomial_2_log_lpmf(1 | log_lambda_psi[s,n], odRN) + bernoulli_lpmf(y2[s,start_idx[n]:end_idx[n]] | r[start_idx[n]:end_idx[n]]));
} else {
lp[1] = neg_binomial_2_log_lpmf(1 | log_lambda_psi[s,n], odRN) + bernoulli_lpmf(y2[s,start_idx[n]:end_idx[n]] | r[start_idx[n]:end_idx[n]]);
}
for (k in 2:n_max[n,s]){
lp[k] = neg_binomial_2_log_lpmf(k | log_lambda_psi[s,n], odRN) + bernoulli_lpmf(y2[s,start_idx[n]:end_idx[n]] | 1-(1-r[start_idx[n]:end_idx[n]])^k);
}
lp_site[s,n] = log_sum_exp(lp);
} else {
lp_site[s, n] = 0;
}
}
}
}
model {
for(s in 1:S) {
beta_psi[s] ~ normal(0, 3); // prior for intercept in poisson model
bioregion_sd[s] ~ normal(0,2);
bioregion_raw[s,] ~ normal(0,1);
to_vector(eps_raw[s,,]) ~ std_normal();
grp_sd[s] ~ normal(0, 1);
zeta[s,] ~ normal(0, 2);
// od
od_mu[s] ~ normal(0,1);
// evc re
evc_raw[s,] ~ normal(0,1);
}
evc_sd ~ normal(0,1);
odRNmu ~ normal(0,1);
beta_trans_det ~ normal(0, 2); // beta for transect detection
beta_det ~ normal(0, 4); // prior for sigma
activ ~ beta(bshape, bscale); //informative prior
log_theta ~ normal(0,2); // prior for theta
for(n in 1:n_site) {
for(j in 1:n_gs) {
for(s in 1:S) {
target += neg_binomial_2_lpmf(n_obs[n,j,s] | lambda[s,n,j], od[s]);
}
y[n,,j] ~ multinomial_logit(to_vector(log_p_raw[n,,j]));
}
for(s in 1:S) {
target += lp_site[s, n];
}
}
}
generated quantities {
array[S, n_site,n_gs] real n_obs_pred;
array[S, n_site, n_gs] real n_obs_true;
array[S, n_site] real N_site;
array[S, n_site] real N_site_pred;
array[n_site, max_int_dist+1] real DetCurve;
array[n_site, n_gs] real log_lik1;
array[S, n_site, n_gs] real log_lik2;
array[n_site, n_gs] real log_lik2_site;
array[S, n_site] real log_lik2_species;
array[n_site] real log_lik;
array[n_site] real log_lik_det;
//array[S, n_site] real Site_lambda;
real av_gs[S];
array[S] simplex[n_gs] eps_gs_ave;
array[S, npc] real pred;
//array[S, npc] real pred_trunc;
array[S, np_reg] real Nhat_reg;
// array[np_reg] real Nhat_reg_design;
real Nhat[S];
//real Nhat_trunc[S];
real Nhat_sum;
int trunc_counter[S];
trunc_counter[S] = 0;
for(s in 1:S) {
eps_gs_ave[s] = exp(zeta[s])/sum(exp(zeta[s]));
av_gs[s] = sum(gs .* eps_gs_ave[s]); //average group size
}
for(n in 1:n_site) {
for(j in 1:n_gs) {
log_lik1[n,j] = multinomial_logit_lpmf(y[n,,j] | to_vector(log_p_raw[n,,j])); //for loo
for(s in 1:S) {
log_lik2[s,n,j] = neg_binomial_2_lpmf(n_obs[n,j,s] | lambda[s,n,j], od[s]); //for loo
n_obs_true[s, n, j] = gs[j] * (neg_binomial_2_log_rng(log_lambda_psi[s,n] + log(eps_ngs[s,n,j]), od[s]));
n_obs_pred[s, n,j] = gs[j] * neg_binomial_2_rng(lambda[s,n,j], od[s]);
}
log_lik2_site[n, j] = log_sum_exp(log_lik2[,n,j]);
}
// get loglik on a site level
log_lik_det[n] = log_sum_exp(log_lik1[n,]);
log_lik[n] = log_sum_exp(log_sum_exp(log_lik_det[n],
log_sum_exp(log_lik2_site[n,])), log_sum_exp(lp_site[,n]));
for(s in 1:S) {
//Site_lambda[s,n] = exp(log_lambda_psi[s,n]);
log_lik2_species[s, n] = log_sum_exp(log_sum_exp(log_lik2[s,n,]), lp_site[s,n]);
N_site[s,n] = sum(n_obs_true[s,n,]);
N_site_pred[s,n] = sum(n_obs_pred[s,n,]);
}
// loop over distance bins
if(keyfun == 0) {
for(j in 0:max_int_dist) { // get DS predictions for distance 0 to max bin distance
DetCurve[n, j+1] = exp(-(j+0.5)^2/(2*sigma[n]^2)); // half normal
}
} else if(keyfun == 1) {
for(j in 0:max_int_dist) { // get DS predictions for distance 0 to max bin distance
DetCurve[n, j+1] = 1 - exp(-((j+0.5)/sigma[n])^(-theta)); //hazard rate
}
}
}
for(s in 1:S) {
for(i in 1:np_reg) Nhat_reg[s,i] = 0;
for(i in 1:npc) {
pred[s,i] = neg_binomial_2_log_rng(X_pred_psi[i,] * beta_psi[s] + eps_bioregion[s, pred_bioreg[i]] + eps_evc[s, pred_evc[i]], od[s]) * prop_pred[i] * av_gs[s]; //offset
if(pred[s,i] > max(N_site[s,])) {
//pred_trunc[s,i] = max(N_site[s,]);
trunc_counter[s] += 1;
}
// for Hog Deer limit to Gippsland region
if(pred_reg[i] != 2 && s == 4) {
pred[s,i] = 0;
}
Nhat_reg[s,pred_reg[i]] += pred[s,i];
}
Nhat[s] = sum(pred[s,]);
//Nhat_trunc[s] = sum(pred_trunc[s,]);
}
Nhat_sum = sum(Nhat);
}Show code
model_negbin_ms <- cmdstan_model(here::here("stan", "count_det_nondet_negbin_ms.stan"))
model_negbin_ms_evc <- cmdstan_model(here::here("stan", "count_det_nondet_negbin_ms_evc.stan"))Fit models
Detection Models
Using the distance package we compare several detection functions (hazard and half-normal) with the inclusion of a parameter (herbaceous understorey cover), which may impact detection rates. The top model (as chosen by AIC) from this selection will be included in our full STAN model.
Show code
filter_behaviour <- TRUE
snapshot_interval <- 2
# Camera information
theta <- 40 * pi / 180 # camera angle in radians
### Get camera information for the camera sites
### For cameras that have problems you need to use the date from when the problem started
cams_curated2 <- dplyr::tbl(con, dbplyr::in_schema("camtrap", "curated_camtrap_operation")) %>%
dplyr::filter(ProjectShortName %in% !!project_short_name) %>% # Only retrieve cam records for hog deer 2023
dplyr::collect() %>%
dplyr::mutate(DateTimeDeploy = as.POSIXct(DateTimeDeploy),
DateTimeRetrieve = as.POSIXct(DateTimeRetrieve),
Tk = as.numeric(DateTimeRetrieve - DateTimeDeploy, units = "secs"), #seconds
Tk_prob = dplyr::coalesce(as.numeric(as.POSIXct(Problem1_to,
format = "%Y-%m-%d %H:%M:%OS") -
as.POSIXct(Problem1_from, format = "%Y-%m-%d %H:%M:%OS"),
units = "secs"), 0),
Tk_adj = Tk-Tk_prob,
Tkt = Tk_adj / snapshot_interval, # snapshot moments: every second second
Effort = (Tk_adj*theta)/(snapshot_interval * pi * 2)) %>%
dplyr::arrange(SiteID)
### Get the deer records (hog deer)
camtrap_records_deer <- dplyr::tbl(con, dbplyr::in_schema(schema = "camtrap", table = "curated_camtrap_records")) %>%
dplyr::filter(scientific_name %in% deer_species_all & ProjectShortName %in% !!project_short_name) %>%
dplyr::select(Species = scientific_name, SiteID, Distance = metadata_Distance, size = metadata_Multiples, Date, Time, Behaviour = metadata_Behaviour) %>%
dplyr::collect() %>%
dplyr::mutate(Distance = dplyr::case_when(Distance == "NA" | is.na(Distance) ~ "999",
TRUE ~ Distance)) %>%
dplyr::mutate(Time = as.POSIXct(Time, format = "%H:%M:%OS"),
Time_n = as.numeric(Time, units = "secs"),
Behaviour = na_if(x = Behaviour, y = "NA")) %>%
dplyr::rowwise() %>%
dplyr::mutate(DistanceMod = list(stringr::str_split(Distance, pattern = "_&_")[[1]])) %>%
dplyr::mutate(Distance = DistanceMod[which.min(as.numeric(stringr::str_extract(DistanceMod, pattern = "[0-9]+")))]) %>%
dplyr::mutate(Distance = dplyr::case_when(Distance == "0-2.5" ~ "0 - 2.5",
Distance == "999" ~ NA_character_,
TRUE ~ Distance)) %>%
dplyr::ungroup() %>%
dplyr::filter(Time_n %% snapshot_interval == 0) %>% #& #snapshot moment interval of 2s
{if(filter_behaviour) dplyr::filter(., is.na(.data[["Behaviour"]])) else .} %>% # filter out behaviors such as camera or marker interaction
dplyr::group_by(SiteID, Time_n, Species) %>%
dplyr::slice(1) %>% # if two photos occur in one second take only one (snapshot moment = 2)
dplyr::ungroup()
### Tidy format for the records
dcount<- camtrap_records_deer %>%
dplyr::select(Species, SiteID, Distance, size, Date, Time, Behaviour) %>%
dplyr::full_join(cams_curated2 %>%
dplyr::select(SiteID, Tkt, Effort), by="SiteID") %>%
dplyr::mutate(object=1:nrow(.)) %>%
dplyr::mutate(size = if_else(is.na(size),0L, size)) %>%
dplyr::arrange(SiteID)
### Format data in a time format for availability
summarised_count <- dcount
summarised_count$Distance <- factor(summarised_count$Distance, levels = c("0 - 2.5", "2.5 - 5", "5 - 7.5", "7.5 - 10", "10+"))
summarised_count <- summarised_count %>%
mutate(distance = case_when(Distance == "0 - 2.5" ~ 1.25,
Distance == "2.5 - 5" ~ 3.75,
Distance == "5 - 7.5" ~ 6.25,
Distance == "7.5 - 10" ~ 8.75,
Distance == "10+" ~ 11.25)) %>%
filter(size == 1)
# site variables
site_vars <- cams_curated2 %>%
left_join(dplyr::tbl(con, dbplyr::in_schema("deervic", "curated_site_data")) %>%
dplyr::filter(SiteID %in% !!c(cams_curated2$SiteID, "47191")) %>%
dplyr::collect() %>%
dplyr::mutate(HerbaceousUnderstoryCover = scale(NNWHUCover + ENWHUCover),
WoodyUnderstoryCover = scale(NWUCover + EWUCover),
Trail = factor(Trail),
SiteID = dplyr::case_when(SiteID == "47191" & CameraID == "HO04101053" ~ "47191A",
SiteID == "47191" & CameraID != "HO04101053" ~ "47191B",
TRUE ~ SiteID))) %>% # native + exotic herbaceous cover
dplyr::arrange(SiteID) %>%
as.data.frame() %>%
select(-Time, -Tkt, -Date, -Effort)
ds_data <- summarised_count %>% left_join(site_vars) %>%
filter(!is.na(HerbaceousUnderstoryCover))
mybreaks <- c(0,2.5,5,7.5,10,12.5)
trunc.list <- list(left=0, right=12.5)
conversion <- convert_units("metre", "metre", "metre")
hn0 <- ds(ds_data, transect = "point", key="hn", adjustment = NULL,
convert_units = conversion, cutpoints = mybreaks, truncation = trunc.list)
hr0 <- ds(ds_data, transect = "point", key="hr", adjustment = NULL,
convert_units = conversion, cutpoints = mybreaks, truncation = trunc.list)
hn1 <- ds(ds_data, transect = "point", key="hn", adjustment = NULL,
formula = ~HerbaceousUnderstoryCover,
convert_units = conversion, cutpoints = mybreaks, truncation = trunc.list)
hr1 <- ds(ds_data, transect = "point", key="hr", adjustment = NULL,
formula = ~HerbaceousUnderstoryCover,
convert_units = conversion, cutpoints = mybreaks, truncation = trunc.list)
kableExtra::kbl(summarize_ds_models(hn0, hr0, hn1, hr1, output = "latex") %>%
mutate(Model = stringr::str_extract(Model, pattern = '(?<=\\{)[^\\}]+')),
digits=3,
caption="Model selection table for deer detection", format = "html") %>%
kable_styling(bootstrap_options = "striped")| Model | Key function | Formula | \(\chi^2\) \(p\)-value | \(\hat{P_a}\) | se(\(\hat{P_a}\)) | \(\Delta\)AIC | |
|---|---|---|---|---|---|---|---|
| 4 | hr1 | Hazard-rate | ~HerbaceousUnderstoryCover | 0 | 0.230 | 0.008 | 0.000 |
| 2 | hr0 | Hazard-rate | ~1 | 0 | 0.228 | 0.008 | 3.759 |
| 1 | hn0 | Half-normal | ~1 | 0 | 0.336 | 0.004 | 186.719 |
| 3 | hn1 | Half-normal | ~HerbaceousUnderstoryCover | 0 | 0.336 | 0.004 | 187.700 |
Model Options
Below we list the models we are implementing, all have a hazard detection function, but we compare four different abundance formulas as well as the inclusion of the EVC random effect.
Show code
#Models to run
models_to_run <- data.frame(formula = c("ab_formula_1",
"ab_formula_2",
"ab_formula_3",
"ab_formula_4",
"ab_formula_3",
"ab_formula_4",
"ab_formula_1",
"ab_formula_2"),
keyfun = c(1,1,1,1,1,1,1,1),
evc = c(F,F,F,F,T,T,T,T))Full Models
We can fit models using cmdstanr. Here we fit models using a poisson distribution. The models we compare are:
- Model with several key abundance covariates and half-normal detection function
- Model with key abundance covariates and hazard-rate detection function
- Null-model (bioregion only) with half-normal detection function
- Model with key abundance covariates, bioregion random-effect and an EVC group random effect.
Show code
model_fits <- list()
for(i in 1:nrow(models_to_run)) {
if(models_to_run$evc[i]) {
model_to_fit <- model_negbin_ms_evc
} else {
model_to_fit <- model_negbin_ms
}
form_to_use <- get(models_to_run$formula[i])
params_to_use <- c(TRUE, labels(terms(ab_formula_1)) %in% labels(terms(form_to_use)))
data_to_use <- multispecies_data
data_to_use$X_psi <- as.matrix(data_to_use$X_psi[,params_to_use])
data_to_use$X_pred_psi <- as.matrix(data_to_use$X_pred_psi[,params_to_use])
data_to_use$m_psi <- sum(params_to_use)
data_to_use$keyfun <- models_to_run$keyfun[i]
model_fits[[i]] <- model_to_fit$sample(data = data_to_use,
chains = nc,
parallel_chains = nc,
init = 0.1,
max_treedepth = 10,
refresh = 25,
show_messages = TRUE,
show_exceptions = FALSE,
save_warmup = FALSE,
iter_sampling = ni,
iter_warmup = nw)
model_fits[[i]]$save_object(paste0("outputs/models/fit_multispecies_",
i,".rds"))
}Model Evaluation
Our strategy for model evaluation is to determine the best performing model out of a limited set of options with different fixed-effects, random-effects and detection functions.
LOO (Leave-One-Out Cross-Validation).
We use leave-one-out cross-validation (LOO-CV); a Bayesian model evaluation process (Vehtari, Gelman, and Gabry 2017; Vehtari et al. 2020) to compare the models. Better performing models according to loo::loo() will have higher elpd values.
Show code
loos <- list()
for(i in 1:length(model_fits)) {
loos[[i]] <- model_fits[[i]]$loo(variables = "log_lik2_species", cores = 8)
}
names(loos) <- paste("model", 1:length(model_fits))
loo_compare_table <- loo::loo_compare(x = loos)
# Plot loo table
gt(loo_compare_table %>%
as.data.frame() %>%
tibble::rownames_to_column()) %>%
fmt_number(decimals = 2) %>%
tab_style(locations = cells_row_groups(), style = cell_text(weight = "bold"))| elpd_diff | se_diff | elpd_loo | se_elpd_loo | p_loo | se_p_loo | looic | se_looic | |
|---|---|---|---|---|---|---|---|---|
| model 7 | 0.00 | 0.00 | 2,120.29 | 7.66 | 1.24 | 0.06 | −4,240.59 | 15.32 |
| model 8 | −0.09 | 0.30 | 2,120.20 | 7.64 | 1.19 | 0.05 | −4,240.41 | 15.29 |
| model 5 | −0.20 | 0.32 | 2,120.09 | 7.64 | 1.17 | 0.05 | −4,240.19 | 15.27 |
| model 6 | −1.76 | 0.68 | 2,118.53 | 7.66 | 1.04 | 0.05 | −4,237.06 | 15.32 |
| model 1 | −1.94 | 0.62 | 2,118.35 | 7.65 | 1.04 | 0.05 | −4,236.70 | 15.29 |
| model 2 | −2.16 | 0.68 | 2,118.13 | 7.65 | 1.00 | 0.05 | −4,236.26 | 15.30 |
| model 3 | −2.40 | 0.68 | 2,117.89 | 7.64 | 0.98 | 0.05 | −4,235.79 | 15.28 |
| model 4 | −4.37 | 0.92 | 2,115.93 | 7.69 | 0.85 | 0.04 | −4,231.86 | 15.38 |
Given that models are similar in there predictive performance at sampled sites we also evaluate the performance of predictions across unsampled sites. Below we see that more complex models (usually having a slighly better elpd), have poorer performance to unsampled areas and lead to unrealistic predictions.
Show code
Nhat_CV <- list()
for(i in 1:length(model_fits)) {
Nhat_CV[[i]] <- model_fits[[i]]$summary("Nhat", CV = ~ sd(.x)/mean(.x)) %>%
mutate(Model = i, Species = deer_species_all)
}
Nhat_CV_c <- bind_rows(Nhat_CV)
Nhat_CV_c %>%
select(Model, Species, CV) %>%
kbl("html", escape=FALSE, caption = "Coefficients of Variation for Total Abundance estimates for the models assessed in the analysis") %>%
kable_styling() %>%
kableExtra::column_spec(3, background = ifelse(Nhat_CV_c$CV > 0.5, "red", ifelse(Nhat_CV_c$CV > 0.2, "orange", "green")), color = "white") %>%
collapse_rows(1) | Model | Species | CV |
|---|---|---|
| 1 | Cervus unicolor | 0.1886410 |
| Dama dama | 0.9422867 | |
| Cervus elaphus | 3.4679094 | |
| Axis porcinus | 0.6694140 | |
| 2 | Cervus unicolor | 0.1595965 |
| Dama dama | 0.4671161 | |
| Cervus elaphus | 1.1285972 | |
| Axis porcinus | 0.5351523 | |
| 3 | Cervus unicolor | 0.1589921 |
| Dama dama | 0.4140389 | |
| Cervus elaphus | 0.8865843 | |
| Axis porcinus | 0.5135488 | |
| 4 | Cervus unicolor | 0.1516583 |
| Dama dama | 0.2145688 | |
| Cervus elaphus | 0.6266303 | |
| Axis porcinus | 0.3045017 | |
| 5 | Cervus unicolor | 0.2132886 |
| Dama dama | 0.5446794 | |
| Cervus elaphus | 3.3479264 | |
| Axis porcinus | 0.7870094 | |
| 6 | Cervus unicolor | 0.2062165 |
| Dama dama | 0.2894896 | |
| Cervus elaphus | 1.5069138 | |
| Axis porcinus | 0.4926303 | |
| 7 | Cervus unicolor | 0.2586611 |
| Dama dama | 0.5460739 | |
| Cervus elaphus | 4.9932521 | |
| Axis porcinus | 0.9147719 | |
| 8 | Cervus unicolor | 0.2183291 |
| Dama dama | 0.4281970 | |
| Cervus elaphus | 4.1957471 | |
| Axis porcinus | 1.0076050 |
Selecting the ‘top’ model
Based on the findings of loo and comparisons of other checks (posterior predictive checks, variation in posterior draws, plausibility of predictions), we see that all model perform similarly well using loo. However, several models produce predictions with lower levels of variation in the posterior predictions. Based on this, we use model 3 hereafter for analysis. This model has fixed effects of scale(sqrt(BareSoil)), scale(sqrt(Nitrogen)), scale(log(PastureDistance)), scale(BIO15) and scale(sqrt(ForestEdge)). It also has a random effect of bioregion for each species and an over-dispersion parameter in the camera counts based on bioregion. This model does not have as many fixed effects as models 1 and 2, and also does not have a second random effect of EVC alongside bioregion (model 6). Unfortunately, the additional random effect of EVC produces much more variable and over-dispersed results for Red Deer, so we opt to use model 3 instead.
Show code
# assign index for top model
top <- 3Posterior predictive checks
Posterior predictive checks allow us to compare the observed data to the model-generated data. For each species we undertake posterior predictive checks for summary statistics relating to the number of deer seen on the cameras at each site. Ideally a well-fit model is able to make predictions that match the observed data. Here counts are the number of snapshot moments with deer. The summary statistics we use for the posterior predictive checks are:
- Maximum counts of deer seen at a site on a camera
- Mean number of deer seen at a site on a camera
- Standard deviation of the counts of deer on the cameras
- Average (mean) counts at sites in a scatter plot
- Proportion of sites with zero counts (camera or transect)
- Observed vs expected proportion of counts
Sambar Deer
Based on the plot below, we see that the PPCs for Sambar perform sufficiently well.
Show code
q95 <- function(x) quantile(x, 0.95, na.rm = T)
# q25 <- function(x) quantile(x, 0.25, na.rm = T)
q75 <- function(x) quantile(x, 0.75, na.rm = T)
sd_90 <- function (x, na.rm = FALSE) {
quants <- quantile(x, c(0.05, 0.95), na.rm = T)
x <- x[x < quants[2] & x > quants[1]]
sqrt(var(if (is.vector(x) || is.factor(x)) x else as.double(x),
na.rm = na.rm))
}
bayes_theme <- c(delwp_cols_seq$Teal[6:8], delwp_cols_shades$Navy[1:3])
bayesplot::color_scheme_set(bayes_theme)
funs <- c(max, mean, sd, ppc_scatter_avg, prop_zero, ppc_dens_overlay)
titles <- c("Max", "Mean", "SD", "Average Site Counts", "Proportion Zeros", "Density of Counts")
ppc_sambar <- list()
# funs <- c(prop_zero, mean, q90, sd)
for(i in 1:length(funs)) {
ppc_sambar[[i]] <- posterior_checks_multispecies(model = model_fits[[top]],
model_data = multispecies_data, species_index = 1,
stat = funs[[i]], integrated = T, only_det = F,
title = titles[i])
}
cowplot::plot_grid(plotlist = ppc_sambar, labels = "AUTO", ncol = 2)
Figure 1: Posterior predictive checks for Sambar Deer
Fallow Deer
The PPC’s below show that fallow has more dispersion in camera counts than Sambar. While the observed counts fall within the intervals of our model, there is higher uncertainty. Some of these findings appear to be driven by a site with very high counts (over 2000 snapshot moments of deer individuals), this is more than twice the second highest.
Show code
ppc_fallow <- list()
for(i in 1:length(funs)) {
ppc_fallow[[i]] <- posterior_checks_multispecies(model = model_fits[[top]],
model_data = multispecies_data, species_index = 2,
stat = funs[[i]], integrated = T, only_det = F,
title = titles[i])
}
cowplot::plot_grid(plotlist = ppc_fallow, labels = "AUTO", ncol = 2)
Figure 2: Posterior predictive checks for Fallow Deer
Red Deer
Red Deer have PPC’s that show a high congruence between predicted and expected counts.
Show code
Figure 3: Posterior predictive checks for Red Deer
Hog Deer
Hog Deer have PPC’s that show our model performs sufficiently well in explaining the patterns of observed counts.
Show code
Figure 4: Posterior predictive checks for Hog Deer
Model Convergence
We observed no divergences or any STAN sampling warnings/issues for our models. To visualise the convergence of the model we can observe the mixing of chains for key parameters below:
beta_det: Parameters for the distance sampling model (intercept and herbaceous understorey)
beta_psi: Parameters for the fixed effect abundance model
bioregion_sd: Parameter for the bioregion random effect standard deviation
odRN: Parameter for the over-dispersion in the Royle-Nichols negative binomial model
od_sdandod_sd: Parameters for the mean and standard deviation of over-dispersion link with bioregion for the camera counts.
Show code
bayesplot::color_scheme_set(unname(delwp_cols[1:6]))
convergence_params <- model_fits[[top]]$draws(c("beta_det",
"beta_psi",
"bioregion_sd",
"beta_trans_det",
"odRN",
"od_mu"))
mcmc_trace(convergence_params, facet_args = list(ncol = 4)) +
theme(panel.spacing = unit(0.5, "lines"))
Figure 5: Chain mixing of several key parameters relating to detection, abundance and dispersion
Model Predictions
Within the STAN model we generate predictions for sampled and un-sampled locations. This provides us with site-level abundance estimates as well as estimates across all (un-sampled) public forest.
Site-based Predictions
Site-based Detections Map
Below we display a detection/non-detection map for all sites.
Show code
vic_regions <- vicmap_query("open-data-platform:delwp_region") %>%
collect() %>%
st_transform(3111) %>%
st_simplify(dTolerance = 500)
delwp_pal <- colorRampPalette(c("#B9C600",
delwp_cols[["Teal"]],
delwp_cols[["Navy"]]))
site_detection_plot <- function(data, regions, cams_curated, species_index) {
cam_data <- cams_curated %>% dplyr::select(SiteID)
cam_data$Detected <- factor(data$any_seen[species_index, ])
cam_data$cam_seen <- factor(as.integer(as.logical(rowSums(data$n_obs[,,species_index]))))
if(species_index == 4) {
regions <- regions %>% filter(delwp_region == "GIPPSLAND")
cam_data <- cam_data %>% st_filter(regions %>% st_transform(4283))
}
lvls <- length(unique(cam_data$Detected, na.rm = T))
plot <- ggplot2::ggplot(data = cam_data) +
ggplot2::geom_sf(data = regions, alpha = 0.75, fill = "grey80") +
ggplot2::geom_sf(aes(fill = Detected), shape = 21, size = 2, alpha = 0.75) +
ggplot2::scale_fill_manual(values = delwp_pal(lvls),
na.value = "transparent", na.translate = F,
name = "Detected", guide = guide_legend(override.aes = list(size = 6))) +
# ggtitle(species_names[species_index])+
theme_bw() +
theme(legend.text = element_text(size = 18), legend.key.size = unit(1, "cm"),
title = element_text(size = 20))
return(plot)
}
detection_plot <- cowplot::plot_grid(site_detection_plot(multispecies_data, regions = vic_regions, cams_curated = cams_curated, species_index = 1),
site_detection_plot(multispecies_data, regions = vic_regions, cams_curated = cams_curated, species_index = 2),
site_detection_plot(multispecies_data, regions = vic_regions, cams_curated = cams_curated, species_index = 3),
site_detection_plot(multispecies_data, regions = vic_regions, cams_curated = cams_curated, species_index = 4), ncol = 2,
labels = c("A) Sambar Deer", "B) Fallow Deer", "C) Red Deer", "D) Hog Deer"), greedy = TRUE)
ggsave(plot = detection_plot, filename = "outputs/plots/all_detection_plot.png", width = 4800, height = 3200, units = "px", bg = "white")
detection_plot
Figure 6: Detections of deer (camera or transects) from across Victoria. Detections of Hog Deer are restricted to Gippsland.
Site-based Prediction Map
Below we plot visualisations of point-estimates for the various deer species surveyed for in this study.
Show code
site_preds <- function(model, cams_curated, species_index) {
rn_dens <- model$summary("N_site", prob_occ = ~ mean(. > 0), trimmed_mean = ~mean(., trim = 0.025), CV = ~sd(.)/mean(.), posterior::default_summary_measures(), posterior::default_convergence_measures())
which_sp <- which(stringr::str_detect(string = rn_dens$variable,
pattern = paste0("N_site\\[",
species_index)))
density_at_sites_rn <- cbind(cams_curated, rn_dens[which_sp, ])
return(density_at_sites_rn)
}
site_density_plot <- function(densities, regions, species) {
densities$density <- cut(densities$mean,
breaks = c(0, 0.25, 0.5, 1, 3, 5, 10, max(densities$mean)),
labels = c("< 0.25",
"0.25 - 0.5",
"0.5 - 1",
"1 - 3",
"3 - 5",
"5 - 10",
"10 +"), include.lowest = T, right = T)
if(species == "Hog") {
regions <- regions %>% filter(delwp_region == "GIPPSLAND")
densities <- densities %>% st_filter(regions %>% st_transform(4283))
}
lvls <- length(unique(densities$density, na.rm = T))
plot <- ggplot2::ggplot(data = densities) +
ggplot2::geom_sf(data = regions, alpha = 0.75, fill = "grey80") +
ggplot2::geom_sf(aes(fill = density, alpha = mean), shape = 21, size = 3) +
ggplot2::scale_fill_manual(values = delwp_pal(lvls),
na.value = "transparent", na.translate = F,
name = "", guide = guide_legend(override.aes = list(size = 6))) +
scale_alpha_continuous(range = c(0.5,1), guide = "none") +
labs(title = paste0('Abundance of ',species ,' Deer'), fill = bquote('Deer per'~km^2)) +
theme_bw() +
theme(legend.text = element_text(size = 18), legend.key.size = unit(1, "cm"),
title = element_text(size = 20))
return(plot)
}
sambar_preds <- site_preds(model_fits[[top]], cams_curated = cams_curated, species_index = 1)
fallow_preds <- site_preds(model_fits[[top]], cams_curated = cams_curated, species_index = 2)
red_preds <- site_preds(model_fits[[top]], cams_curated = cams_curated, species_index = 3)
hog_preds <- site_preds(model_fits[[top]], cams_curated = cams_curated, species_index = 4)
site_deer_predictions <- list(Sambar = sambar_preds,
Fallow = fallow_preds,
Red = red_preds,
Hog = hog_preds)
saveRDS(site_deer_predictions, "outputs/site_deer_predictions.rds")
cowplot::plot_grid(site_density_plot(sambar_preds, vic_regions, species = "Sambar"),
site_density_plot(fallow_preds, vic_regions, species = "Fallow"),
site_density_plot(red_preds, vic_regions, species = "Red"),
site_density_plot(hog_preds, vic_regions, species = "Hog"), ncol = 1)
Figure 7: Site-level density estimates for all sites sampled as part of statewide and hog deer surveys. Point-density estimates are from trimmed means of the posterior draws (trimming of the top and bottom 1% of draws)
Average density estimates varied across sites for all four species. Sambar deer were the most commonly detected species on the camera traps (104/317 sites), and density estimates at sites ranged from 0 to 28.31 deer/km2, for fallow, fewer site detections (30), but more variability in site density gave a range between 0 and 23.25 deer/km2. Red deer were only detected on 12 sites with densities ranging from 0 to 6.81. Finally, Hog Deer, which were only detected on 22 cameras in South Gippsland, had site densities ranging from 0 to 16.9 deer/km2.
Site-density Summaries
As a sanity-check we compare the average modelled densities at sites with (i) no evidence of deer, (ii) evidence of deer present on transects, and (iii) evidence of deer present on cameras. We expect that average densities are generally higher at sites that detected some form of deer than sites that did not detect any sign of deer. Additionally, we would also expect average densities to be generally higher at sites that had detections on cameras than those with only detections from transects. The table below shows these expectations to be correct.
Show code
density_summary_table <- function(preds, model_data, species, species_index) {
cam_seen <- as.integer(as.logical(rowSums(model_data$n_obs[,,species_index])))
transect_seen <- model_data$transects %>%
filter(Species == !!deer_species_all[species_index]) %>%
mutate(NoCam = case_when(Survey == "Camera" ~ 0L,
TRUE ~ 1)) %>%
group_by(SiteID) %>%
summarise(TransectDet = max(NoCam*Presence)) %>%
pull(TransectDet)
preds_sum <- preds %>%
st_drop_geometry() %>%
mutate(`Species` = species,
`CameraDetection` = cam_seen,
`AnyDetection` = model_data$any_seen[species_index, ],
`OnlyTransect` = transect_seen,
`Detection` = case_when(CameraDetection == 0 & AnyDetection == 0 ~ "Not seen",
CameraDetection == 0 & AnyDetection == 1 ~ "Only detected on transects",
CameraDetection == 1 & OnlyTransect == 0 ~ "Only seen on cameras",
CameraDetection == 1 & OnlyTransect == 1 ~ "Seen on both camera and transect")) %>%
group_by(`Species`, `Detection`) %>%
summarise(`Number of Sites` = n(),
`Average Density` = mean(trimmed_mean)) %>%
ungroup()
return(preds_sum)
}
density_summary <- bind_rows(
density_summary_table(sambar_preds, multispecies_data,
species = "Sambar", species_index = 1),
density_summary_table(fallow_preds, multispecies_data,
species = "Fallow", species_index = 2),
density_summary_table(red_preds, multispecies_data,
species = "Red", species_index = 3),
density_summary_table(hog_preds, multispecies_data,
species = "Hog", species_index = 4))
density_summary %>%
kbl(format = "html", digits = 2, caption = "Average (98% trimmed mean) density estimates at the various sites groups of sites") %>%
kable_styling("striped") %>%
collapse_rows(1)| Species | Detection | Number of Sites | Average Density |
|---|---|---|---|
| Sambar | Not seen | 182 | 0.50 |
| Only detected on transects | 31 | 1.54 | |
| Only seen on cameras | 50 | 2.20 | |
| Seen on both camera and transect | 54 | 2.70 | |
| Fallow | Not seen | 259 | 0.10 |
| Only detected on transects | 28 | 0.18 | |
| Only seen on cameras | 20 | 0.35 | |
| Seen on both camera and transect | 10 | 0.52 | |
| Red | Not seen | 304 | 0.01 |
| Only detected on transects | 1 | 0.51 | |
| Only seen on cameras | 5 | 0.54 | |
| Seen on both camera and transect | 7 | 0.82 | |
| Hog | Not seen | 293 | 0.12 |
| Only detected on transects | 2 | 1.04 | |
| Only seen on cameras | 22 | 1.49 |
Regional and Statewide Abundance
Within our model we calculate abundance/density for deer in each of the 7.457^{4} km2 of public land. Based on these spatial predictions we can estimate abundance at a regional level (6 DEECA regions) and across the whole state.
Statewide Maps
Using the model predictions (“pred”) for all suitable public land we generate a raster (1km2 resolution). We save the average spatial estimates under outputs/rasters (tif files) and also provide binned plots (png files) in outputs/plots.
Show code
pred_raster_full <- terra::rast(prediction_raster)
pred_raster <- terra::app(pred_raster_full[[stringr::str_subset(
stringr::str_remove_all(labels(terms(ab_formula_1)),
"scale[(]|[)]|log[(]|sqrt[(]"),
pattern = "[*]", negate = T)]], mean)
mean_raster <- list()
occ_raster <- list()
mean_raster_discrete <- list()
# gp_preds_draws_all <- model_fits[[top]]$draws("pred", format = "matrix")
gp_preds_draws_all <- model_fits[[top]]$draws("pred", format = "matrix")
# gp_preds_summary_all <- model_fits[[top]]$summary("pred", prob_occ = ~ mean(. > 0))
for(i in 1:length(deer_species_all)) {
which_sp <- which(stringr::str_detect(string = colnames(gp_preds_draws_all),
pattern = paste0("pred\\[", i)))
gp_preds_draws_sp <- gp_preds_draws_all[,which_sp]
terra::values(pred_raster)[!is.na(terra::values(pred_raster))] <- apply(gp_preds_draws_sp,
2,
mean,
na.rm = T,
trim = 0.001) #filter out highest draw
mean_raster[[i]] <- pred_raster
mean_raster_discrete[[i]] <- mean_raster[[i]]
max_pred <- max(values(mean_raster[[i]]), na.rm = T)
values(mean_raster_discrete[[i]]) <- cut(values(mean_raster_discrete[[i]]) ,
breaks = c(0,0.25,0.5,1,3,5,10,max_pred),
include.lowest = T, right = T,
labels = c("< 0.25",
"0.25 - 0.5",
"0.5 - 1",
"1 - 3",
"3 - 5",
"5 - 10",
"10 +"))
occ_raster[[i]] <- pred_raster
terra::values(occ_raster[[i]])[!is.na(terra::values(occ_raster[[i]]))] <- apply(gp_preds_draws_sp, 2, function(x) sum(x > 0)/length(x))
}
# combine mean rasters together
combined_raster <- rast(mean_raster)
names(combined_raster) <- c("Average Sambar Deer Density (per km2)",
"Average Fallow Deer Density (per km2)",
"Average Red Deer Density (per km2)",
"Average Hog Deer Density (per km2)")
combined_occupancy_raster <- rast(occ_raster)
names(combined_occupancy_raster) <- c("Average Sambar Deer Occupancy Probability",
"Average Fallow Deer Occupancy Probability",
"Average Red Deer Occupancy Probability",
"Average Hog Deer Occupancy Probability")
writeRaster(combined_occupancy_raster, "outputs/rasters/combined_occupancy_raster.tif", overwrite = T)
writeRaster(combined_raster, "outputs/rasters/combined_deer_average_density.tif", overwrite = T)Show code
# reg <- VicmapR::vicmap_query("open-data-platform:delwp_region") %>%
# VicmapR::collect() %>%
# sf::st_transform(3111) %>%
# sf::st_simplify(dTolerance = 250)
state <- VicmapR::vicmap_query("open-data-platform:vmlite_victoria_polygon_su5") %>%
filter(state != "NSW" & state != "SA" & feature_type_code != "sea") %>%
VicmapR::collect() %>%
sf::st_transform(3111)
gippsland <- vic_regions %>% filter(delwp_region == "GIPPSLAND")
plot_abundance <- function(raster, state, species, crop = NULL) {
if(!is.null(crop)) {
raster <- terra::crop(raster, vect(crop), mask = T)
state <- crop
}
lvls <- length(unique(values(raster, na.rm = T)))
plot <- ggplot2::ggplot() +
ggplot2::geom_sf(data = state, alpha = 1, linewidth = 0.5, fill = "grey90") +
tidyterra::geom_spatraster(data = raster, na.rm = T) +
# tidyterra::scale_fill_terrain_d(na.translate = FALSE) +
ggplot2::scale_fill_manual(values = delwp_pal(lvls), na.value = "transparent", na.translate = F) +
ggplot2::labs(fill = bquote('Deer per'~km^2), title = paste0("Average Abundance of ", species, " on Victorian Public Land")) +
ggspatial::annotation_scale() +
delwp_theme()
return(plot)
}
sambar_abundance_plot <- plot_abundance(mean_raster_discrete[[1]],
state = state,
species = "Sambar Deer")
fallow_abundance_plot <- plot_abundance(mean_raster_discrete[[2]],
state = state,
species = "Fallow Deer")
red_abundance_plot <- plot_abundance(mean_raster_discrete[[3]],
state = state,
species = "Red Deer")
hog_abundance_plot <- plot_abundance(mean_raster_discrete[[4]],
state = state, crop = gippsland,
species = "Hog Deer")
ggsave(plot = sambar_abundance_plot, filename = "outputs/plots/sambar_abundance_plot.png",
width = 2400, height = 1600, units = "px")
ggsave(plot = fallow_abundance_plot, filename = "outputs/plots/fallow_abundance_plot.png",
width = 2400, height = 1600, units = "px")
ggsave(plot = red_abundance_plot, filename = "outputs/plots/red_abundance_plot.png",
width = 2400, height = 1600, units = "px")
ggsave(plot = hog_abundance_plot, filename = "outputs/plots/hog_abundance_plot.png",
width = 2400, height = 1600, units = "px")
ggsave(plot = cowplot::plot_grid(sambar_abundance_plot, fallow_abundance_plot, red_abundance_plot, hog_abundance_plot, ncol = 2, labels = "AUTO", greedy = F), filename = "outputs/plots/all_abundance_plot.png", width = 4800, height = 3200, units = "px", bg = "white")
cowplot::plot_grid(sambar_abundance_plot, fallow_abundance_plot, red_abundance_plot, hog_abundance_plot, ncol = 1)
Figure 8: Abundance of Sambar, Fallow, Red and Hog Deer across Victoria, dark-grey area reflects area not included in predictions (i.e. not public land)
Regional Abundance Estimates
For each DEECA region we provide species-level estimates of abundance with 90 % confidence interval. We also calculate the model-based average density within each region based on the abundance and the total area of public land within each region.
Show code
Nhat_sum <- model_fits[[top]]$summary("Nhat")
format_nhat <- function(sp, data = Nhat_sum) {
fd <- data %>%
filter(variable == !!paste0("Nhat[", sp, "]"))
paste0("$\\hat N$ = ",
formatC(fd$mean, digits = 0, big.mark = ",", format = "f"),
" [90% CI: ",
formatC(fd$q5, digits = 0, big.mark = ",", format = "f"),
" – ",
formatC(fd$q95, digits = 0, big.mark = ",", format = "f"),
"]")
}
reg <- VicmapR::vicmap_query("open-data-platform:delwp_region") %>%
VicmapR::collect() %>%
dplyr::group_by(delwp_region) %>%
dplyr::summarise(geometry = sf::st_combine(geometry)) %>%
dplyr::ungroup() %>%
dplyr::mutate(delwp_region_fact = as.integer(factor(delwp_region)))
abundance_table <- function(model, regions, pred_area, caption, species_index, return_data = F) {
reg <- model$summary("Nhat_reg", prob_occ = ~ mean(. > 0), trimmed_mean = ~mean(., trim = 0.025), posterior::default_summary_measures(), posterior::default_convergence_measures())
which_sp <- which(stringr::str_detect(string = reg$variable,
pattern = paste0("Nhat_reg\\[", species_index)))
regional_abundance <- reg[which_sp,] %>%
dplyr::bind_rows(model$summary("Nhat", prob_occ = ~ mean(. > 0), trimmed_mean = ~mean(., trim = 0.025), posterior::default_summary_measures(), posterior::default_convergence_measures())[species_index,]) %>%
dplyr::mutate(variable = c(regions$delwp_region, "TOTAL"),
`Area km2` = round(c(pred_area, sum(pred_area))),
`Average Density (per km2)` = round(median/`Area km2`, 2)) %>%
dplyr::transmute(Region = variable,
`Mean` = round(mean),
Median = round(median),
SD = round(sd),
MAD = round(mad),
`5 %` = round(q5),
`95 %` = round(q95),
`Area km2`,
`Average Density (per km2)`)
if(return_data) {
if(species_index == 4) {
regional_abundance <- regional_abundance %>%
filter(Region == "GIPPSLAND")
}
return(regional_abundance %>%
mutate(Species = species_names[species_index], across(2:8, ~ formatC(.x, digits = 0, format = "f", big.mark = ",") )) %>%
select(Species, everything()))
}
kableExtra::kbl(regional_abundance, digits = 2, format = "html", caption = caption, format.args = list(big.mark = ",")) %>%
kableExtra::kable_styling("striped") %>%
kableExtra::column_spec(1, bold = TRUE) %>%
kableExtra::row_spec(6, hline_after = T) %>%
kableExtra::row_spec(7, background = "#A5A1B5", color = "black", bold = T, hline_after = T)
}
pred_a_df <- data.frame(pred_reg = multispecies_data$pred_reg,
area = multispecies_data$prop_pred) %>%
group_by(pred_reg) %>%
summarise(area = sum(area))
# Save Abundance Table
all_ab_tab <- bind_rows(abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, species_index = 1, return_data = T),
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, species_index = 2, return_data = T),
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, species_index = 3, return_data = T),
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, species_index = 4, return_data = T))
write.csv(all_ab_tab, "outputs/tables/regional_abundance_estimates.csv")
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, caption = "Regional estimates of Sambar Deer Density", species_index = 1)| Region | Mean | Median | SD | MAD | 5 % | 95 % | Area km2 | Average Density (per km2) |
|---|---|---|---|---|---|---|---|---|
| BARWON SOUTH WEST | 166 | 107 | 277 | 72 | 35 | 430 | 4,745 | 0.02 |
| GIPPSLAND | 55,110 | 54,109 | 8,898 | 8,675 | 42,618 | 70,948 | 25,211 | 2.15 |
| GRAMPIANS | 1,726 | 1,461 | 2,032 | 648 | 716 | 3,266 | 9,896 | 0.15 |
| HUME | 60,133 | 58,676 | 12,360 | 11,614 | 43,334 | 82,188 | 16,890 | 3.47 |
| LODDON MALLEE | 1,254 | 940 | 1,761 | 429 | 460 | 2,589 | 15,597 | 0.06 |
| PORT PHILLIP | 5,247 | 5,041 | 1,334 | 1,258 | 3,367 | 7,694 | 2,231 | 2.26 |
| TOTAL | 123,636 | 121,200 | 19,657 | 18,946 | 96,200 | 157,638 | 74,570 | 1.63 |
Show code
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, caption = "Regional estimates of Fallow Deer Density", species_index = 2)| Region | Mean | Median | SD | MAD | 5 % | 95 % | Area km2 | Average Density (per km2) |
|---|---|---|---|---|---|---|---|---|
| BARWON SOUTH WEST | 11,928 | 8,146 | 14,091 | 5,142 | 3,049 | 31,646 | 4,745 | 1.72 |
| GIPPSLAND | 12,619 | 11,592 | 5,001 | 3,711 | 6,914 | 21,794 | 25,211 | 0.46 |
| GRAMPIANS | 4,622 | 3,865 | 3,203 | 1,529 | 2,106 | 9,264 | 9,896 | 0.39 |
| HUME | 16,288 | 15,179 | 5,519 | 4,475 | 9,362 | 26,721 | 16,890 | 0.90 |
| LODDON MALLEE | 3,467 | 2,922 | 2,198 | 1,380 | 1,384 | 7,277 | 15,597 | 0.19 |
| PORT PHILLIP | 1,554 | 1,378 | 767 | 533 | 761 | 2,879 | 2,231 | 0.62 |
| TOTAL | 50,478 | 45,609 | 20,900 | 13,463 | 29,888 | 85,063 | 74,570 | 0.61 |
Show code
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, caption = "Regional estimates of Red Deer Density", species_index = 3)| Region | Mean | Median | SD | MAD | 5 % | 95 % | Area km2 | Average Density (per km2) |
|---|---|---|---|---|---|---|---|---|
| BARWON SOUTH WEST | 6,481 | 4,063 | 8,018 | 2,684 | 1,559 | 19,399 | 4,745 | 0.86 |
| GIPPSLAND | 197 | 143 | 196 | 92 | 49 | 502 | 25,211 | 0.01 |
| GRAMPIANS | 5,705 | 4,093 | 5,709 | 2,228 | 1,705 | 14,695 | 9,896 | 0.41 |
| HUME | 1,175 | 971 | 739 | 530 | 420 | 2,638 | 16,890 | 0.06 |
| LODDON MALLEE | 248 | 24 | 1,524 | 30 | 1 | 794 | 15,597 | 0.00 |
| PORT PHILLIP | 40 | 24 | 83 | 18 | 5 | 103 | 2,231 | 0.01 |
| TOTAL | 13,845 | 10,135 | 12,275 | 4,967 | 4,719 | 35,465 | 74,570 | 0.14 |
Show code
abundance_table(model_fits[[top]], regions = reg, pred_area = pred_a_df$area, caption = "Regional estimates of Hog Deer Density", species_index = 4) | Region | Mean | Median | SD | MAD | 5 % | 95 % | Area km2 | Average Density (per km2) |
|---|---|---|---|---|---|---|---|---|
| BARWON SOUTH WEST | 0 | 0 | 0 | 0 | 0 | 0 | 4,745 | 0.00 |
| GIPPSLAND | 4,408 | 3,884 | 2,263 | 1,529 | 2,121 | 8,464 | 25,211 | 0.15 |
| GRAMPIANS | 0 | 0 | 0 | 0 | 0 | 0 | 9,896 | 0.00 |
| HUME | 0 | 0 | 0 | 0 | 0 | 0 | 16,890 | 0.00 |
| LODDON MALLEE | 0 | 0 | 0 | 0 | 0 | 0 | 15,597 | 0.00 |
| PORT PHILLIP | 0 | 0 | 0 | 0 | 0 | 0 | 2,231 | 0.00 |
| TOTAL | 4,408 | 3,884 | 2,263 | 1,529 | 2,121 | 8,464 | 74,570 | 0.05 |
The total estimated number of Sambar deer in Victoria was estimated at \(\hat N\) = 123,636 [90% CI: 96,200 – 157,638], followed by Fallow deer at \(\hat N\) = 50,478 [90% CI: 29,888 – 85,063], Red deer (\(\hat N\) = 13,845 [90% CI: 4,719 – 35,465]), and Hog Deer (\(\hat N\) = 4,408 [90% CI: 2,121 – 8,464]).
Model Covariates
Our model uses covariates to inform three separate sub-models:
- A distance-sampling detection process in front of the camera
- An imperfect detection/observation model based on the transect methods and the camera survey
- An abundance model based on spatially-derived covariates, as well as a bioregion random effect
The following section explores the effect of these covariates and sub-models on detection and abundance.
Detection Processes
At a site, there are two processes that may lead us to not record deer on a camera when in reality they occupy the site (whereby the site is the home range area surrounding the camera.) Firstly, deer may be available for detection and enter the sampling area in front of the camera. However, due to a function of their distance from the camera we fail to detect this individual. Secondly, they may occupy a site (known from transect signs) but never enter the camera field of view, in this case we estimate the various detection rates from the various methods. The detection rate of the camera in this method is regarded as the spatial availability of deer for camera trap sampling.
Distance-sampling
Below we show the average detection rate for a given group of deer in front of the camera (up to 12.5m).
Show code
det_rates <- model_fits[[top]]$summary("p") %>%
mutate(var = stringr::str_extract(variable, "(?<=\\[).+?(?=\\])")) %>%
separate(var, into = c("site", "Group Size"), sep = ",")
av_det_rates <- det_rates %>%
group_by(`Group Size`) %>%
summarise(mean = mean(mean)) %>%
ungroup() %>%
transmute(`Group Size`,
`Average Site Detection Probability` = mean)
av_det_rates %>%
bind_rows() %>%
arrange(`Group Size`) %>%
kbl("html", digits = 3, caption = "Average detection rates for different group sizes") %>%
kable_styling("striped") %>%
collapse_rows(1)| Group Size | Average Site Detection Probability |
|---|---|
| 1 | 0.297 |
| 2 | 0.379 |
| 3 | 0.439 |
| 4 | 0.485 |
| 5 | 0.523 |
We can generate detection function plots that show how detection rates fall-off over varying distances. Initial exploration showed that there may be some relationship between herbaceous understorey and detection probability for Sambar Deer. However, when pooled together we see that herbaceous understorey appears to have no effect on detection.
Show code
site_vars <- dplyr::tbl(con, dbplyr::in_schema("deervic", "curated_site_data")) %>%
dplyr::filter(SiteID %in% !!c(cams_curated$SiteID, "47191")) %>%
dplyr::collect() %>%
dplyr::mutate(HerbaceousUnderstoryCover = NNWHUCover + ENWHUCover,
SiteID = dplyr::case_when(SiteID == "47191" & CameraID == "HO04101053" ~ "47191A",
SiteID == "47191" & CameraID != "HO04101053" ~ "47191B",
TRUE ~ SiteID)) %>% # native + exotic herbaceous cover
dplyr::arrange(SiteID) %>%
as.data.frame()
n_distance_bins <- 5
delta <- 2.5
midpts <- c(1.25, 3.75, 6.25, 8.75, 11.25)
max_distance <- 12.5
gs <- 1:5
pa <- matrix(data = NA, nrow = max(gs), ncol = length(midpts))
for(j in 1:max(gs)) {
pa[j, ] <- sapply(midpts, function(x) {pr_closest(x-(0.5*delta),
x+(0.5*delta),
max_distance = max_distance,
n = j)})
}
det_curve <- model_fits[[top]]$draws("DetCurve", format = "draws_matrix") %>%
as.data.frame() %>%
# head(250) %>%
pivot_longer(cols = everything())
det_curve_wr <- det_curve %>%
mutate(var = stringr::str_extract(name, "(?<=\\[).+?(?=\\])")) %>%
separate(var, into = c("s", "Distance"), sep = ",") %>%
mutate(Distance = as.numeric(Distance)-0.5)
det_vars_pred <- site_vars %>%
mutate(s = as.character(1:nrow(.)),
herbaceouslvl = cut(HerbaceousUnderstoryCover,
breaks = c(0, 25, 50, 75, 105),
labels = c("0 - 25%",
"25 - 50%",
"50 - 75%",
"75 - 100%"),
include.lowest = T, right = FALSE))
det_curve_sum <- det_curve_wr %>%
mutate(Distance = as.numeric(Distance)) %>%
left_join(det_vars_pred) %>%
group_by(herbaceouslvl, Distance) %>%
summarise(median = quantile(value, 0.5),
q5 = quantile(value, 0.05),
q95 = quantile(value, 0.95))
scaleHN <- det_curve_sum %>% filter(Distance == 1.5) %>% pull(median) %>% mean() # Approx scaling
y_comb_list <- list()
for(i in 1:max(gs)) {
y_comb_list[[i]] <- colSums(multispecies_data$y[,,i]) %>%
as.data.frame() %>%
`colnames<-`("Count") %>%
mutate(Distance = midpts,
Bin = 1:5,
gs = i,
Prop = Count/(max(Count)),
CountS = Count/pa[Bin,i])
}
y_combined <- bind_rows(y_comb_list) %>%
group_by(Distance) %>%
mutate(SumCount = sum(Count)) %>%
ungroup() %>%
mutate(propC = Count/SumCount,
PropSM = (CountS/max(CountS)) * propC) %>%
group_by(Distance) %>%
summarise(PropS = sum(PropSM) * scaleHN)
detection_plot_HN <- ggplot(aes(x = Distance), data = det_curve_sum) +
geom_col(aes(y = PropS), fill = "grey70", colour = "grey20", width = 2.5, data = y_combined, alpha = 0.7) +
# geom_error bar(aes(ymin = PropSq5, ymax = PropSq95), data = y_combined) +
# geom_line(aes(y = HNS)) +
geom_ribbon(aes(ymin = q5, ymax = q95, fill = herbaceouslvl), alpha = 0.5) +
geom_line(aes(y = median, colour = herbaceouslvl)) +
scale_x_continuous(expand = c(0, 0)) +
scale_y_continuous(expand = c(0, 0), limits = c(0,1)) +
scale_fill_manual(values = unname(delwp_cols[c(1,2,3,8)])) +
scale_colour_manual(values = unname(delwp_cols[c(1,2,3,8)])) +
ylab("Detection probability (p)") +
labs(fill = "Herbaceous\nUnderstorey\nCover", colour = "Herbaceous\nUnderstorey\nCover") +
theme_classic()
detection_plot_HN
Figure 9: Distance-sampling detection process for Deer (group size = 1-5). Herbaceous understorey cover in the area surrounding appears to have some impact on detection probability
Transect-surveys
For Sambar, Fallow and Red Deer we were able to estimate detection rates of the various methods used to detect deer (camera, footprints, pellets, rubbings and wallows).
Show code
all_draws <- model_fits[[top]]$draws("beta_trans_det", format = "df")
# det_marginal_effects <- list()
# det_plot <- list()
obs_vars <- unlist(stringr::str_remove_all(string = labels(terms(transect_formula)), pattern = "log|scale|\\)|\\("))
obs_cols <- unlist(stringr::str_remove_all(string = colnames(multispecies_data$trans_pred_matrix), pattern = "log|scale|\\)|\\("))
obs_logs <- unlist(stringr::str_detect(string = colnames(multispecies_data$trans_pred_matrix), pattern = "log\\("))
obs_labs <- c("Survey Method")
fac <- c(TRUE, TRUE, TRUE, TRUE, TRUE)
fac2 <- c(TRUE)
det_obs <- multispecies_data$transects %>%
mutate(Survey = factor(Survey))
det_marginal_effects <- list()
det_plot <- list()
params_w_levs <- levels(det_obs$Survey)
for(i in 1:(length(obs_cols))) {
det_marginal_effects[[i]] <- marginal_effects_cmd(all_draws,
param = "beta_trans_det",
param_number = i, log = obs_logs[i],
model_data = det_obs,
model_column = obs_vars[c(1,attr(multispecies_data$trans_pred_matrix, "assign")[-1])[i]],
transition = FALSE) %>%
mutate(group = param,
param = params_w_levs[i],
factor = fac[i],
variable = as.numeric(variable))
if(fac[i]) {
det_marginal_effects[[i]] <- det_marginal_effects[[i]]
}
}
marginal_prob <- function(x, pwr = 3) {
xm <- 1-x
return(1-(xm^pwr))
}
det_marginal_effects_bind <- bind_rows(det_marginal_effects) %>%
rowwise() %>%
mutate(value = case_when(param != "Camera" ~ marginal_prob(value),
TRUE ~ value))
det_marginal_effects_split <- split(det_marginal_effects_bind, det_marginal_effects_bind$group)
for(i in 1:length(det_marginal_effects_split)) {
if(fac2[i]) {
plot_data <- det_marginal_effects_split[[i]] %>%
mutate(variable = param,
param = group)
} else {
plot_data <- det_marginal_effects_split[[i]]
}
det_plot[[i]] <- plot_data
}
data_for_plot <- bind_rows(det_plot) %>%
mutate(species = "All (combined)") %>%
mutate(Density = "1 deer/km^2^")
data_d3 <- data_for_plot %>%
mutate(value = 1 - (1 - value)^3,
Density = "3 deer/km^2^")
data_d5 <- data_for_plot %>%
mutate(value = 1 - (1 - value)^5,
Density = "5 deer/km^2^")
data_combined <- bind_rows(data_for_plot, data_d3, data_d5)
marginal_effects_plot_cmd_all(data_combined,
factor = TRUE,
ylab = "Detection probability") +
xlab("Survey method") +
scale_y_continuous(limits = c(0,1)) +
theme(legend.position = "top") +
scale_fill_manual(values = unname(delwp_cols[c(1,2,3)])) +
# scale_x_discrete(expand = c(0, 0)) +
theme_classic() +
theme(legend.text = ggtext::element_markdown())
Figure 10: Unconditional detection probabilities for the various methods of survey, across various deer densities. Camera trap detection probability is based on average deployment length (53 days), while transects are based on 3 transect searches
Abundance Processes
Bioregion
We used a spatially-derived random-effect of the bioregion of the sampled site. This random-effect allows us the ability to make predictions that include the variance associated with this random effect. This random-effect also minimises predictions in areas without deer detection’s (e.g. the mallee).
Show code
bioregion_sd <- model_fits[[top]]$summary("bioregion_sd")
bayesplot::color_scheme_set(bayes_theme)
bioregion_contribution <- function(model, data, species, species_index) {
eps_bioregion <- model$draws("eps_bioregion", format = "matrix")
which_sp <- which(stringr::str_detect(string = colnames(eps_bioregion),
pattern = paste0("eps_bioregion\\[", i)))
eps_bioregion <- eps_bioregion[,which_sp]
bioregion_data <- data[["bioreg_sf"]]
colnames(eps_bioregion) <- bioregion_data$bioregion
plot <- mcmc_areas(eps_bioregion, prob_outer = 0.9) +
ggtitle(species)
return(plot)
}
bio_plot_data <- list()
for(i in 1:length(species_names)) {
bio_plot_data[[i]] <- bioregion_contribution(model = model_fits[[top]],
data = multispecies_data,
species = species_names[i], species_index = i)
}
cowplot::plot_grid(plotlist = bio_plot_data, ncol = 1)
Figure 11: Influence of the bioregion random effect on abundance (log-scale).
Bioregion usually had a substantial impact on abundance estimates across the state. The variance (standard deviation) associated with the bioregion random effect was large for Sambar (\(\sigma\) = 3.96, 90% CI: 2.8 - 5.38), but less for Fallow (\(\sigma\) = 1.67, 90% CI: 0.85 - 2.73). Red Deer also exhibited large variance in the bioregion random effect (\(\sigma\) = 3.7, 90% CI: 2.48 - 5.23), as well Hog (\(\sigma\) = 4.4, 90% CI: 3.11 - 5.93). The larger variances for Hog, Red and Sambar reflect their more refined range boundaries, wheras Fallow show less clear range boundaries and thus show less variance across the state.
Show code
# Bioregion map
bioregion_eff <- model_fits[[top]]$summary("eps_bioregion", posterior::default_summary_measures())
bioreg <- VicmapR::vicmap_query("open-data-platform:vbioreg100") %>%
VicmapR::collect() %>%
group_by(bioregion, bioregion_code) %>%
summarise(geometry = st_union(geometry)) %>%
ungroup()
bioregion_nhat <- model_fits[[top]]$summary("Nhat_bioreg", prob_occ = ~ mean(. > 0), trimmed_mean = ~mean(., trim = 0.025), posterior::default_summary_measures())
bioregion_density <- list()
for(i in 1:length(species_names)) {
which_sp <- which(stringr::str_detect(string = bioregion_nhat$variable,
pattern = paste0("Nhat_bioreg\\[", i)))
bioregion_nhat_sp <- bioregion_nhat[which_sp,]
bioregion_nhat_sp$bioregion <- multispecies_data$bioreg_sf$bioregion
bioregion_nhat_sp$Area <- as.numeric(table(multispecies_data$pred_bioreg))
bioregion_nhat_sp <- bioregion_nhat_sp %>%
mutate(`Average Density` = trimmed_mean/Area,
`Density (5%)` = q5/Area,
`Density (95%)` = q95/Area,
`CV` = sd/trimmed_mean,
label = paste0(bioregion, " [D = "
,round(`Average Density`,2),
" (",round(`Density (5%)`,2)," - ",round(`Density (95%)`, 2),")]"))
bioregion_density[[i]] <- left_join(bioreg, bioregion_nhat_sp, by = "bioregion")
}
bioregion_effects <- list()
for(i in 1:length(species_names)) {
which_sp <- which(stringr::str_detect(string = bioregion_eff$variable,
pattern = paste0("eps_bioregion\\[", i)))
bioregion_eff_sp <- bioregion_eff[which_sp,]
intercept <- model_fits[[top]]$summary(paste0("beta_psi[",i,",1]"), mean = ~ mean(.))$mean
bioregion_eff_sp$bioregion <- multispecies_data$bioreg_sf$bioregion
bioregion_eff_sp$Area <- as.numeric(table(multispecies_data$pred_bioreg))
bioregion_eff_sp <- bioregion_eff_sp %>%
mutate(`Average Effect` = mean)
bioregion_effects[[i]] <- left_join(bioreg, bioregion_eff_sp, by = "bioregion")
}
bioregion_ggplot <- function(effect_df, title) {
ggplot(data = effect_df) +
geom_sf(data = state, linewidth = 0.5, fill = "grey90") +
geom_sf(aes(fill = `Average Effect`)) +
geom_label_repel(data = effect_df, aes(label = bioregion, fill = `Average Effect`, geometry = geometry),
size = 2.5, stat = "sf_coordinates", colour = "white") +
scale_fill_gradient2(low = "#B9C600",
mid = delwp_cols[["Teal"]],
high = delwp_cols[["Navy"]], guide = guide_coloursteps(title = "Average Effect\n (log-scale)")) +
labs(x = "Longitude", y = "Latitude") +
ggspatial::annotation_scale() +
delwp_theme()
}
# cl_br <- st_intersection(crown_land, bioreg %>% filter(bioregion == "Bridgewater") %>% st_transform(3111))
cpb <- cowplot::plot_grid(bioregion_ggplot(bioregion_effects[[1]], "Sambar"),
bioregion_ggplot(bioregion_effects[[2]], "Fallow"),
bioregion_ggplot(bioregion_effects[[3]], "Red"),
bioregion_ggplot(bioregion_effects[[4]], "Hog"), labels = c("A) Sambar Deer", "B) Fallow Deer", "C) Red Deer", "D) Hog Deer"), ncol = 2)
ggsave(plot = cpb, filename = "outputs/plots/bioregion_effects_plot.png", width = 4800, height = 3200, units = "px", bg = "white")
cpb
Figure 12: Effect of Victorian Bioregions on abundance of deer
Fixed-parameter Effects
Show code
# beta_summaries
beta_summaries <- model_fits[[top]]$summary("beta_psi")
format_cov <- function(sp, b, data = beta_summaries, variable = "beta_psi") {
fd <- data %>%
filter(variable == !!paste0(variable, "[", sp, ",",b+1,"]"))
paste0("$\\beta$ = ",
round(fd$mean, 2),
" [90% CI: ",
round(fd$q5, 2),
" – ",
round(fd$q95, 2),
"]")
}
ab_joined_list <- list()
for(j in 1:length(species_names)) {
beta_draws <- model_fits[[top]]$draws("beta_psi", format = "df")
# which_sp <- which(stringr::str_detect(string = colnames(beta_draws),
# pattern = paste0("beta_psi\\[", i)))
#
# beta_draws <- beta_draws[,which_sp]
ab_marginal_effects <- list()
ab_plot <- list()
ab_to_use <- ab_formula_3
phi_vars <- unlist(stringr::str_remove_all(string = labels(terms(ab_to_use)), pattern = "log|scale|sqrt|\\)|\\("))
phi_logs <- unlist(stringr::str_detect(string = labels(terms(ab_to_use)), pattern = "log\\("))
phi_sqrts <- c(0.5,0.5,1,1,0.5)
phi_labs <- c("Bare Soil (%)",
"Nitrogen (%)",
"Distance to Pastural Land (m)",
"Precipitation Seasonality",
'Forest Edge per km^2^ (m)')
fac <- c(FALSE, FALSE, FALSE, FALSE, FALSE)
for(i in 1:length(phi_vars)) {
ab_marginal_effects[[i]] <- marginal_effects_cmd(beta_draws,
param = "beta_psi", species_index = j,
param_number = i+1, log = phi_logs[i],
model_data = multispecies_data[["raw_data"]] %>% mutate(Nitrogen = Nitrogen*100),
abundance = TRUE,
pwr = phi_sqrts[i],
model_column = phi_vars[i],
transition = FALSE) %>%
mutate(species = species_names[j])
}
ab_joined_list[[j]] <- bind_rows(ab_marginal_effects)
}
all_me_data <- bind_rows(ab_joined_list) %>%
mutate(param = case_when(param == "BareSoil" ~ "Bare Soil (%)",
param == "Nitrogen" ~ "Nitrogen (%)",
param == "PastureDistance" ~ "Distance to Pastural Land (m)",
param == "BIO15" ~ "Precipitation Seasonality",
param == "ForestEdge" ~ 'Forest Edge per km^2^ (m)'))
cov_plot <- marginal_effects_plot_cmd_all(all_me_data %>% mutate(species = factor(species, levels = species_names),
param = factor(param, levels = phi_labs)),
col = "DarkGreen",
factor = FALSE,
ylab = "Contribution to Abundance (log-scale)") +
ggplot2::facet_grid(species~param, scales = "free") +
delwp_theme() +
theme(strip.text = ggtext::element_markdown()) +
xlab("Covariate Value")
ggsave(plot = cov_plot, filename = "outputs/plots/abundance_covariate_effects.png", width = 3000, height = 2000, units = "px")
cov_plot
Figure 13: Marginal response curves for the various fixed-effect parameters used in the model.
Of the five fixed-effect covariates used in the top model we found that all had tangible effects for one or more species. The direction and magnitude of these effects often varied between species. Bare soil had a slight positive effect on Sambar abundance (\(\beta\) = 0.37 [90% CI: 0.02 – 0.72]), negligible effect on Fallow (\(\beta\) = -0.1 [90% CI: -0.63 – 0.4]) and Red Deer (\(\beta\) = -0.73 [90% CI: -2.01 – 0.54]), and a slight negative effect on Hog Deer abundance (\(\beta\) = -0.72 [90% CI: -1.57 – 0.12]). Nitrogen generally had weaker, more variable effects. The effect of nitrogen was weakly positive for Sambar (\(\beta\) = 0.31 [90% CI: -0.08 – 0.68]), and Hog Deer (\(\beta\) = 0.53 [90% CI: -0.54 – 1.6]); but weakly negative for Fallow (\(\beta\) = 0.22 [90% CI: -0.36 – 0.72]), and Red Deer (\(\beta\) = -0.45 [90% CI: -1.48 – 0.63]). The effect of distance to pasture was consistently negative across all four species, which can also be interpreted as a prefence of deer to be ‘closer’ to pastural areas. However the effect of this covariate varied in magnitude for Sambar (\(\beta\) = -0.53 [90% CI: -0.83 – -0.22]), Fallow (\(\beta\) = -0.89 [90% CI: -1.35 – -0.43]), Red (\(\beta\) = -1.61 [90% CI: -2.71 – -0.48]), and Hog Deer (\(\beta\) = -0.3 [90% CI: -0.8 – 0.22]). Sambar Deer weakly favored areas with less seasonal variation in precipitation (\(\beta\) = -0.23 [90% CI: -0.57 – 0.12]), with Fallow (\(\beta\) = 0.39 [90% CI: 0.02 – 0.8]), Red (\(\beta\) = 1.05 [90% CI: 0.05 – 2.02]), and Hog Deer (\(\beta\) = 0.77 [90% CI: 0.21 – 1.31]) all showing positive relationships between precipitation seasonality and abundance. Lastly, we saw that the amount of forest edge in the landscape positively impacted abundance for Sambar (\(\beta\) = 0.17 [90% CI: -0.06 – 0.4]), and Fallow Deer (\(\beta\) = 0.61 [90% CI: 0.29 – 0.93]). This relationship was weakly negative for Red Deer (\(\beta\) = -0.7 [90% CI: -1.39 – 0.01]), and weakly positive for Hog Deer (\(\beta\) = 0.32 [90% CI: -0.07 – 0.69]).
Additional Parameters of Interest
Below is a table with several model parameters we estimated:
Show code
key_params <- model_fits[[top]]$summary(c("av_gs",
"od_mu",
"odRN",
"activ",
"Nhat_sum"))
key_params %>%
kbl(format = "html", digits = 2, caption = "Several key outputs and parameters of the model not reported earlier") %>%
kable_styling("striped")| variable | mean | median | sd | mad | q5 | q95 | rhat | ess_bulk | ess_tail |
|---|---|---|---|---|---|---|---|---|---|
| av_gs[1] | 1.01 | 1.00 | 0.00 | 0.00 | 1.00 | 1.01 | 1.00 | 1067.27 | 1375.61 |
| av_gs[2] | 1.02 | 1.02 | 0.02 | 0.01 | 1.00 | 1.05 | 1.02 | 564.16 | 775.02 |
| av_gs[3] | 1.04 | 1.03 | 0.04 | 0.02 | 1.01 | 1.10 | 1.00 | 1923.67 | 1447.03 |
| av_gs[4] | 1.03 | 1.01 | 0.04 | 0.01 | 1.00 | 1.10 | 1.00 | 591.52 | 583.61 |
| od_mu[1] | -1.93 | -1.93 | 0.21 | 0.21 | -2.28 | -1.59 | 1.00 | 965.07 | 1255.70 |
| od_mu[2] | -3.62 | -3.61 | 0.27 | 0.26 | -4.07 | -3.18 | 1.01 | 774.65 | 1157.38 |
| od_mu[3] | -2.86 | -2.86 | 0.36 | 0.37 | -3.43 | -2.29 | 1.01 | 1207.00 | 1323.88 |
| od_mu[4] | -2.54 | -2.53 | 0.32 | 0.33 | -3.08 | -2.00 | 1.00 | 913.74 | 916.78 |
| odRN | 0.79 | 0.78 | 0.14 | 0.14 | 0.59 | 1.04 | 1.01 | 815.79 | 1125.54 |
| activ | 0.52 | 0.52 | 0.02 | 0.02 | 0.48 | 0.56 | 1.00 | 2217.73 | 1339.29 |
| Nhat_sum | 192366.95 | 186810.00 | 34001.78 | 30296.93 | 146731.70 | 255490.00 | 1.00 | 1450.13 | 1403.75 |
Relating abundance to vegetation
From our point density estimates of deer, we can model relationships between deer densities at sites and a range of structural vegetation measures collected during the surveys. For this part of the analysis we implement a multivariate regression.
Firstly the response variables we model are:
- Bare Ground Cover (Beta Distribution)
- Native Woody Understorey Cover (Beta Distribution)
- Native Non-woody Understorey Cover (Beta Distribution)
- Presence of seedlings (Negative Binomial Distribution)
- Presence of saplings (Negative Binomial Distribution)
- Presence of exotic flora (Bernoulli Distribution)
While our main predictor of interest is the density of deer at a site, we also control for predictors that are expected to have an impact on vegetation growth and composition. The predictors controlled for include features relating to climate, ecology, topography, fire history and forest structure. They are the following predictors:
- BIO01: Annual Mean Temperature
- BIO04: Temperature Seasonality
- BIO06: Min Temperature of Coldest Month
- BIO12: Annual Precipitation
- BIO15: Precipitation Seasonality (Coefficient of Variation)
- Soil Nitrogen
- Time since last fire (factor grouping)
- TWIND (Topographic wetness index)
- Forest Edge
- Canopy Cover
- Top Height
- EVC Group (random effect with 19 groups)
Data Preparation
We can use the sum of estimated deer density for the four species at each site as our key predictor variable. Given that we already extracted a range of bioclimatic and environmental covariates for the original model, we can combine those with the data available here.
Note that this model is run on a subset of sites (sites surveyed as part of the statewide deer survey), as the hog deer sites did not have complete vegetation metrics recorded.
Show code
density <- model_fits[[top]]$draws("N_site", format = "matrix")
site_estimates <- list()
site_density <- list()
for(i in 1:n_site) {
site_estimates[[i]] <- character()
for(j in 1:length(deer_species_all)) {
site_estimates[[i]][j] <- paste0("N_site[", j, ",", i, "]")
}
site_density[[i]] <- sum(colMeans(density)[site_estimates[[i]]])
}
site_density_ul <- unlist(site_density)
site_density_df <- cams_curated %>%
mutate(AllDeerDensity = site_density_ul,
Bioregion = factor(multispecies_data$site_bioreg),
EVC = factor(multispecies_data$site_evc)) %>%
filter(ProjectShortName == "StatewideDeer") %>%
left_join(multispecies_data[["raw_data"]] %>% select(-EVC, - BIOREGION)) %>%
mutate(TSLF_bin = as.factor(round(TSLF_bin)))
site_vars <- dplyr::tbl(con, dbplyr::in_schema("deervic", "curated_site_data")) %>%
dplyr::filter(SiteID %in% !!c(cams_curated$SiteID, "47191")) %>%
dplyr::collect() %>%
dplyr::mutate(HerbaceousUnderstoryCover = NNWHUCover + ENWHUCover,
Exotics = EWUCover + ENWHUCover,
ExoticPresence = as.integer(as.logical(Exotics)),
SeedlingsOrSaplings = as.integer(as.logical(Seedlings+Saplings)),
SeedlingsPresence = as.integer(as.logical(Seedlings)),
SaplingsPresence = as.integer(as.logical(Saplings)),
SiteID = dplyr::case_when(SiteID == "47191" & CameraID == "HO04101053" ~ "47191A",
SiteID == "47191" & CameraID != "HO04101053" ~ "47191B",
TRUE ~ SiteID)) %>% # native + exotic herbaceous cover
dplyr::arrange(SiteID) %>%
as.data.frame()
site_density_df_joined <- site_density_df %>%
left_join(site_vars) %>%
mutate(BGroundCover = ifelse(BGroundCover == 0, 0.125, BGroundCover)/100,
NWUCover = ifelse(NWUCover == 0, 0.125, NWUCover)/100,
NNWHUCover = ifelse(NNWHUCover == 0, 0.125, NNWHUCover)/100)Model Execution
We use the brms package to run our model in a Bayesian framework (Bürkner 2017, 2018, 2021). Our model is run in parallel on four cores over four chains for 1,000 sampling iterations per chain (1,000 warm up iterations).
Show code
bform1 <- bf(mvbind(BGroundCover,
NWUCover,
NNWHUCover,
Seedlings,
Saplings,
ExoticPresence) ~ scale(sqrt(AllDeerDensity))
+ scale(BIO01)
+ scale(BIO04)
+ scale(BIO06)
+ scale(BIO12)
+ scale(BIO15)
+ scale(Nitrogen)
+ TSLF_bin
+ scale(TWIND)
+ scale(sqrt(ForestEdge))
+ scale(sqrt(CanopyCov))
+ scale(sqrt(TopHeight))
+ (1|EVC))
vegfit <- brm(bform1,
data = site_density_df_joined,
family = list(Beta(link = "logit"),
Beta(link = "logit"),
Beta(link = "logit"),
negbinomial(),
negbinomial(),
bernoulli(link = "logit")),
inits = "0",
chains = 4,
cores = 4,
iter = 2000,
warmup = 1000,
backend = "cmdstanr")
saveRDS(vegfit, "outputs/models/vegfit.rds")Model Results
Model Summary
Below we show a complete summary of the model, including all coefficients estimated for all the parameters and response variables. Standard deviations for the EVC random effect and the estimates of the \(\phi\) parameter in the beta-distribution models, and the shape parameter in the negative binomial models are also shown. Note that all Rhats > 1.05 and all ESS > 200, suggesting good chain mixing and convergence.
Show code
vegfit <- readRDS("outputs/models/vegfit.rds")
veg_coefs <- fixef(vegfit, probs = c(0.05, 0.95), pars = c("BGroundCover_scalesqrtAllDeerDensity",
"NWUCover_scalesqrtAllDeerDensity",
"NNWHUCover_scalesqrtAllDeerDensity",
"Seedlings_scalesqrtAllDeerDensity",
"Saplings_scalesqrtAllDeerDensity",
"ExoticPresence_scalesqrtAllDeerDensity")) %>%
as.data.frame() %>%
mutate(Effect = paste0(c("Bare Ground Cover",
"Native Woody Understorey Cover",
"Native Non-woody Understorey Cover",
"Presence of seedlings",
"Presence of saplings",
"Presence of exotic flora"), " ~ Deer Density"))
print_conditional <- function(x, i, d = 3, OR = FALSE) {
row <- x[i,]
name <- "\\beta"
if(OR) {
row <- row %>%
mutate(across(where(is.numeric), exp))
name <- "OR"
}
paste0("$",name,"$ = ", round(row[["Estimate"]], d), " [90% CI: ", round(row[["Q5"]], d), ", ", round(row[["Q95"]], d), "]")
}
summary(vegfit) Family: MV(beta, beta, beta, negbinomial, negbinomial, bernoulli)
Links: mu = logit; phi = identity
mu = logit; phi = identity
mu = logit; phi = identity
mu = log; shape = identity
mu = log; shape = identity
mu = logit
Formula: BGroundCover ~ scale(sqrt(AllDeerDensity)) + scale(BIO01) + scale(BIO04) + scale(BIO06) + scale(BIO12) + scale(BIO15) + scale(Nitrogen) + TSLF_bin + scale(TWIND) + scale(sqrt(ForestEdge)) + scale(sqrt(CanopyCov)) + scale(sqrt(TopHeight)) + (1 | EVC)
NWUCover ~ scale(sqrt(AllDeerDensity)) + scale(BIO01) + scale(BIO04) + scale(BIO06) + scale(BIO12) + scale(BIO15) + scale(Nitrogen) + TSLF_bin + scale(TWIND) + scale(sqrt(ForestEdge)) + scale(sqrt(CanopyCov)) + scale(sqrt(TopHeight)) + (1 | EVC)
NNWHUCover ~ scale(sqrt(AllDeerDensity)) + scale(BIO01) + scale(BIO04) + scale(BIO06) + scale(BIO12) + scale(BIO15) + scale(Nitrogen) + TSLF_bin + scale(TWIND) + scale(sqrt(ForestEdge)) + scale(sqrt(CanopyCov)) + scale(sqrt(TopHeight)) + (1 | EVC)
Seedlings ~ scale(sqrt(AllDeerDensity)) + scale(BIO01) + scale(BIO04) + scale(BIO06) + scale(BIO12) + scale(BIO15) + scale(Nitrogen) + TSLF_bin + scale(TWIND) + scale(sqrt(ForestEdge)) + scale(sqrt(CanopyCov)) + scale(sqrt(TopHeight)) + (1 | EVC)
Saplings ~ scale(sqrt(AllDeerDensity)) + scale(BIO01) + scale(BIO04) + scale(BIO06) + scale(BIO12) + scale(BIO15) + scale(Nitrogen) + TSLF_bin + scale(TWIND) + scale(sqrt(ForestEdge)) + scale(sqrt(CanopyCov)) + scale(sqrt(TopHeight)) + (1 | EVC)
ExoticPresence ~ scale(sqrt(AllDeerDensity)) + scale(BIO01) + scale(BIO04) + scale(BIO06) + scale(BIO12) + scale(BIO15) + scale(Nitrogen) + TSLF_bin + scale(TWIND) + scale(sqrt(ForestEdge)) + scale(sqrt(CanopyCov)) + scale(sqrt(TopHeight)) + (1 | EVC)
Data: site_density_df_joined (Number of observations: 252)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Group-Level Effects:
~EVC (Number of levels: 19)
Estimate Est.Error l-95% CI u-95% CI
sd(BGroundCover_Intercept) 0.26 0.12 0.04 0.52
sd(NWUCover_Intercept) 0.13 0.09 0.01 0.36
sd(NNWHUCover_Intercept) 0.23 0.13 0.01 0.52
sd(Seedlings_Intercept) 1.21 0.42 0.53 2.17
sd(Saplings_Intercept) 0.84 0.33 0.22 1.54
sd(ExoticPresence_Intercept) 0.54 0.38 0.03 1.47
Rhat Bulk_ESS Tail_ESS
sd(BGroundCover_Intercept) 1.00 1034 776
sd(NWUCover_Intercept) 1.01 1532 1963
sd(NNWHUCover_Intercept) 1.00 839 1274
sd(Seedlings_Intercept) 1.00 910 810
sd(Saplings_Intercept) 1.00 878 864
sd(ExoticPresence_Intercept) 1.00 845 1776
Population-Level Effects:
Estimate Est.Error l-95% CI
BGroundCover_Intercept -2.22 0.21 -2.63
NWUCover_Intercept -0.64 0.17 -0.97
NNWHUCover_Intercept -0.30 0.20 -0.69
Seedlings_Intercept 3.35 0.54 2.30
Saplings_Intercept 3.48 0.47 2.54
ExoticPresence_Intercept -1.35 0.52 -2.39
BGroundCover_scalesqrtAllDeerDensity -0.19 0.09 -0.36
BGroundCover_scaleBIO01 0.31 0.41 -0.51
BGroundCover_scaleBIO04 0.11 0.26 -0.38
BGroundCover_scaleBIO06 -0.17 0.48 -1.08
BGroundCover_scaleBIO12 -0.20 0.14 -0.49
BGroundCover_scaleBIO15 -0.22 0.08 -0.39
BGroundCover_scaleNitrogen -0.03 0.16 -0.34
BGroundCover_TSLF_bin3 0.12 0.22 -0.31
BGroundCover_TSLF_bin4 0.07 0.24 -0.40
BGroundCover_TSLF_bin5 -0.22 0.25 -0.70
BGroundCover_scaleTWIND 0.05 0.11 -0.16
BGroundCover_scalesqrtForestEdge -0.01 0.07 -0.16
BGroundCover_scalesqrtCanopyCov -0.01 0.07 -0.15
BGroundCover_scalesqrtTopHeight -0.26 0.08 -0.41
NWUCover_scalesqrtAllDeerDensity -0.18 0.08 -0.34
NWUCover_scaleBIO01 0.01 0.41 -0.80
NWUCover_scaleBIO04 -0.00 0.25 -0.49
NWUCover_scaleBIO06 -0.00 0.47 -0.91
NWUCover_scaleBIO12 0.23 0.14 -0.05
NWUCover_scaleBIO15 0.18 0.08 0.02
NWUCover_scaleNitrogen -0.02 0.16 -0.35
NWUCover_TSLF_bin3 -0.37 0.20 -0.75
NWUCover_TSLF_bin4 -0.59 0.23 -1.05
NWUCover_TSLF_bin5 -0.93 0.25 -1.41
NWUCover_scaleTWIND -0.04 0.12 -0.27
NWUCover_scalesqrtForestEdge 0.07 0.08 -0.08
NWUCover_scalesqrtCanopyCov -0.05 0.07 -0.19
NWUCover_scalesqrtTopHeight -0.13 0.08 -0.28
NNWHUCover_scalesqrtAllDeerDensity 0.21 0.09 0.04
NNWHUCover_scaleBIO01 0.32 0.49 -0.67
NNWHUCover_scaleBIO04 -0.38 0.30 -0.95
NNWHUCover_scaleBIO06 -0.64 0.56 -1.70
NNWHUCover_scaleBIO12 0.38 0.16 0.05
NNWHUCover_scaleBIO15 0.13 0.09 -0.06
NNWHUCover_scaleNitrogen -0.26 0.18 -0.62
NNWHUCover_TSLF_bin3 -0.29 0.23 -0.73
NNWHUCover_TSLF_bin4 -0.14 0.25 -0.62
NNWHUCover_TSLF_bin5 -0.15 0.26 -0.66
NNWHUCover_scaleTWIND -0.08 0.13 -0.33
NNWHUCover_scalesqrtForestEdge 0.00 0.08 -0.16
NNWHUCover_scalesqrtCanopyCov 0.03 0.08 -0.12
NNWHUCover_scalesqrtTopHeight 0.02 0.08 -0.15
Seedlings_scalesqrtAllDeerDensity 0.29 0.23 -0.16
Seedlings_scaleBIO01 0.16 0.92 -1.70
Seedlings_scaleBIO04 -0.13 0.60 -1.31
Seedlings_scaleBIO06 -0.57 1.02 -2.60
Seedlings_scaleBIO12 0.54 0.45 -0.34
Seedlings_scaleBIO15 -0.16 0.29 -0.73
Seedlings_scaleNitrogen -0.91 0.39 -1.66
Seedlings_TSLF_bin3 -1.14 0.53 -2.19
Seedlings_TSLF_bin4 -1.20 0.53 -2.30
Seedlings_TSLF_bin5 -1.63 0.58 -2.76
Seedlings_scaleTWIND -0.09 0.24 -0.56
Seedlings_scalesqrtForestEdge -0.40 0.17 -0.74
Seedlings_scalesqrtCanopyCov -0.09 0.18 -0.44
Seedlings_scalesqrtTopHeight -0.92 0.20 -1.31
Saplings_scalesqrtAllDeerDensity -0.06 0.18 -0.41
Saplings_scaleBIO01 0.27 0.83 -1.40
Saplings_scaleBIO04 0.08 0.51 -0.89
Saplings_scaleBIO06 0.16 0.91 -1.61
Saplings_scaleBIO12 0.32 0.33 -0.32
Saplings_scaleBIO15 -0.43 0.20 -0.81
Saplings_scaleNitrogen 0.12 0.33 -0.54
Saplings_TSLF_bin3 -0.57 0.47 -1.49
Saplings_TSLF_bin4 -1.76 0.49 -2.69
Saplings_TSLF_bin5 -1.31 0.54 -2.37
Saplings_scaleTWIND 0.09 0.23 -0.35
Saplings_scalesqrtForestEdge -0.31 0.15 -0.61
Saplings_scalesqrtCanopyCov 0.02 0.15 -0.27
Saplings_scalesqrtTopHeight -0.56 0.16 -0.88
ExoticPresence_scalesqrtAllDeerDensity 0.57 0.22 0.16
ExoticPresence_scaleBIO01 0.49 1.10 -1.66
ExoticPresence_scaleBIO04 -0.15 0.70 -1.57
ExoticPresence_scaleBIO06 -0.76 1.30 -3.44
ExoticPresence_scaleBIO12 0.24 0.41 -0.54
ExoticPresence_scaleBIO15 0.39 0.21 -0.02
ExoticPresence_scaleNitrogen -1.18 0.47 -2.14
ExoticPresence_TSLF_bin3 0.65 0.59 -0.51
ExoticPresence_TSLF_bin4 0.69 0.64 -0.56
ExoticPresence_TSLF_bin5 0.98 0.66 -0.30
ExoticPresence_scaleTWIND -0.37 0.30 -0.96
ExoticPresence_scalesqrtForestEdge 0.09 0.19 -0.27
ExoticPresence_scalesqrtCanopyCov 0.25 0.19 -0.11
ExoticPresence_scalesqrtTopHeight 0.09 0.20 -0.31
u-95% CI Rhat Bulk_ESS
BGroundCover_Intercept -1.82 1.00 2158
NWUCover_Intercept -0.31 1.00 2559
NNWHUCover_Intercept 0.08 1.00 2571
Seedlings_Intercept 4.39 1.00 1279
Saplings_Intercept 4.37 1.00 2005
ExoticPresence_Intercept -0.36 1.00 2582
BGroundCover_scalesqrtAllDeerDensity -0.02 1.00 4687
BGroundCover_scaleBIO01 1.11 1.00 1883
BGroundCover_scaleBIO04 0.64 1.00 1903
BGroundCover_scaleBIO06 0.79 1.00 1907
BGroundCover_scaleBIO12 0.08 1.00 4359
BGroundCover_scaleBIO15 -0.06 1.00 2815
BGroundCover_scaleNitrogen 0.27 1.00 3683
BGroundCover_TSLF_bin3 0.56 1.00 2248
BGroundCover_TSLF_bin4 0.55 1.00 2217
BGroundCover_TSLF_bin5 0.28 1.00 2346
BGroundCover_scaleTWIND 0.26 1.00 4924
BGroundCover_scalesqrtForestEdge 0.13 1.00 4597
BGroundCover_scalesqrtCanopyCov 0.13 1.00 4717
BGroundCover_scalesqrtTopHeight -0.10 1.00 5845
NWUCover_scalesqrtAllDeerDensity -0.03 1.00 5675
NWUCover_scaleBIO01 0.80 1.00 1875
NWUCover_scaleBIO04 0.50 1.00 1946
NWUCover_scaleBIO06 0.91 1.00 1870
NWUCover_scaleBIO12 0.51 1.00 3733
NWUCover_scaleBIO15 0.34 1.00 3929
NWUCover_scaleNitrogen 0.30 1.00 4368
NWUCover_TSLF_bin3 -0.00 1.00 2829
NWUCover_TSLF_bin4 -0.16 1.00 2754
NWUCover_TSLF_bin5 -0.45 1.00 2820
NWUCover_scaleTWIND 0.19 1.00 4556
NWUCover_scalesqrtForestEdge 0.22 1.00 5637
NWUCover_scalesqrtCanopyCov 0.09 1.00 5162
NWUCover_scalesqrtTopHeight 0.02 1.00 4765
NNWHUCover_scalesqrtAllDeerDensity 0.38 1.00 4804
NNWHUCover_scaleBIO01 1.24 1.00 1233
NNWHUCover_scaleBIO04 0.23 1.00 1307
NNWHUCover_scaleBIO06 0.48 1.00 1255
NNWHUCover_scaleBIO12 0.70 1.00 3533
NNWHUCover_scaleBIO15 0.31 1.00 3821
NNWHUCover_scaleNitrogen 0.08 1.00 3856
NNWHUCover_TSLF_bin3 0.16 1.00 2378
NNWHUCover_TSLF_bin4 0.36 1.00 2271
NNWHUCover_TSLF_bin5 0.36 1.00 2183
NNWHUCover_scaleTWIND 0.18 1.00 4158
NNWHUCover_scalesqrtForestEdge 0.16 1.00 5358
NNWHUCover_scalesqrtCanopyCov 0.18 1.00 4879
NNWHUCover_scalesqrtTopHeight 0.19 1.00 3543
Seedlings_scalesqrtAllDeerDensity 0.73 1.00 2830
Seedlings_scaleBIO01 2.00 1.00 1278
Seedlings_scaleBIO04 1.03 1.00 1415
Seedlings_scaleBIO06 1.43 1.00 1344
Seedlings_scaleBIO12 1.44 1.00 1632
Seedlings_scaleBIO15 0.42 1.00 2246
Seedlings_scaleNitrogen -0.12 1.00 3566
Seedlings_TSLF_bin3 -0.10 1.00 2490
Seedlings_TSLF_bin4 -0.18 1.00 2044
Seedlings_TSLF_bin5 -0.52 1.00 2428
Seedlings_scaleTWIND 0.39 1.00 3517
Seedlings_scalesqrtForestEdge -0.05 1.00 3970
Seedlings_scalesqrtCanopyCov 0.27 1.00 5056
Seedlings_scalesqrtTopHeight -0.53 1.00 4813
Saplings_scalesqrtAllDeerDensity 0.30 1.00 3807
Saplings_scaleBIO01 1.88 1.00 1739
Saplings_scaleBIO04 1.12 1.00 1819
Saplings_scaleBIO06 1.98 1.00 1751
Saplings_scaleBIO12 0.95 1.00 2468
Saplings_scaleBIO15 -0.04 1.00 2812
Saplings_scaleNitrogen 0.76 1.00 4379
Saplings_TSLF_bin3 0.37 1.00 2372
Saplings_TSLF_bin4 -0.78 1.00 2398
Saplings_TSLF_bin5 -0.20 1.00 2281
Saplings_scaleTWIND 0.52 1.00 3962
Saplings_scalesqrtForestEdge -0.00 1.00 4540
Saplings_scalesqrtCanopyCov 0.30 1.00 4530
Saplings_scalesqrtTopHeight -0.24 1.00 3521
ExoticPresence_scalesqrtAllDeerDensity 1.03 1.00 4130
ExoticPresence_scaleBIO01 2.72 1.00 1710
ExoticPresence_scaleBIO04 1.18 1.00 1742
ExoticPresence_scaleBIO06 1.71 1.00 1661
ExoticPresence_scaleBIO12 1.06 1.00 3609
ExoticPresence_scaleBIO15 0.80 1.00 3582
ExoticPresence_scaleNitrogen -0.30 1.00 3576
ExoticPresence_TSLF_bin3 1.73 1.00 2447
ExoticPresence_TSLF_bin4 1.95 1.00 2298
ExoticPresence_TSLF_bin5 2.25 1.00 2429
ExoticPresence_scaleTWIND 0.22 1.00 4915
ExoticPresence_scalesqrtForestEdge 0.46 1.00 5594
ExoticPresence_scalesqrtCanopyCov 0.62 1.00 5763
ExoticPresence_scalesqrtTopHeight 0.49 1.00 4473
Tail_ESS
BGroundCover_Intercept 3153
NWUCover_Intercept 2995
NNWHUCover_Intercept 2773
Seedlings_Intercept 1905
Saplings_Intercept 2636
ExoticPresence_Intercept 2705
BGroundCover_scalesqrtAllDeerDensity 3230
BGroundCover_scaleBIO01 2372
BGroundCover_scaleBIO04 2455
BGroundCover_scaleBIO06 2234
BGroundCover_scaleBIO12 3140
BGroundCover_scaleBIO15 2941
BGroundCover_scaleNitrogen 3239
BGroundCover_TSLF_bin3 3086
BGroundCover_TSLF_bin4 2645
BGroundCover_TSLF_bin5 2799
BGroundCover_scaleTWIND 3445
BGroundCover_scalesqrtForestEdge 3330
BGroundCover_scalesqrtCanopyCov 3252
BGroundCover_scalesqrtTopHeight 2652
NWUCover_scalesqrtAllDeerDensity 3164
NWUCover_scaleBIO01 2524
NWUCover_scaleBIO04 2571
NWUCover_scaleBIO06 2484
NWUCover_scaleBIO12 3163
NWUCover_scaleBIO15 3415
NWUCover_scaleNitrogen 3701
NWUCover_TSLF_bin3 3246
NWUCover_TSLF_bin4 3484
NWUCover_TSLF_bin5 3336
NWUCover_scaleTWIND 3492
NWUCover_scalesqrtForestEdge 3346
NWUCover_scalesqrtCanopyCov 3327
NWUCover_scalesqrtTopHeight 3090
NNWHUCover_scalesqrtAllDeerDensity 3291
NNWHUCover_scaleBIO01 2318
NNWHUCover_scaleBIO04 2446
NNWHUCover_scaleBIO06 2517
NNWHUCover_scaleBIO12 3006
NNWHUCover_scaleBIO15 3329
NNWHUCover_scaleNitrogen 3248
NNWHUCover_TSLF_bin3 2799
NNWHUCover_TSLF_bin4 3055
NNWHUCover_TSLF_bin5 2710
NNWHUCover_scaleTWIND 3157
NNWHUCover_scalesqrtForestEdge 3192
NNWHUCover_scalesqrtCanopyCov 3064
NNWHUCover_scalesqrtTopHeight 2625
Seedlings_scalesqrtAllDeerDensity 1744
Seedlings_scaleBIO01 1551
Seedlings_scaleBIO04 2246
Seedlings_scaleBIO06 1938
Seedlings_scaleBIO12 1525
Seedlings_scaleBIO15 3026
Seedlings_scaleNitrogen 2843
Seedlings_TSLF_bin3 3032
Seedlings_TSLF_bin4 2443
Seedlings_TSLF_bin5 2276
Seedlings_scaleTWIND 3303
Seedlings_scalesqrtForestEdge 3157
Seedlings_scalesqrtCanopyCov 3131
Seedlings_scalesqrtTopHeight 2737
Saplings_scalesqrtAllDeerDensity 3059
Saplings_scaleBIO01 2275
Saplings_scaleBIO04 2145
Saplings_scaleBIO06 2364
Saplings_scaleBIO12 2618
Saplings_scaleBIO15 3090
Saplings_scaleNitrogen 3185
Saplings_TSLF_bin3 2657
Saplings_TSLF_bin4 2648
Saplings_TSLF_bin5 3068
Saplings_scaleTWIND 2882
Saplings_scalesqrtForestEdge 3309
Saplings_scalesqrtCanopyCov 3151
Saplings_scalesqrtTopHeight 2758
ExoticPresence_scalesqrtAllDeerDensity 2991
ExoticPresence_scaleBIO01 2203
ExoticPresence_scaleBIO04 2199
ExoticPresence_scaleBIO06 2117
ExoticPresence_scaleBIO12 3246
ExoticPresence_scaleBIO15 3497
ExoticPresence_scaleNitrogen 3283
ExoticPresence_TSLF_bin3 2534
ExoticPresence_TSLF_bin4 2964
ExoticPresence_TSLF_bin5 2689
ExoticPresence_scaleTWIND 3660
ExoticPresence_scalesqrtForestEdge 3723
ExoticPresence_scalesqrtCanopyCov 3331
ExoticPresence_scalesqrtTopHeight 2899
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
phi_BGroundCover 6.87 0.73 5.55 8.43 1.00 4708
phi_NWUCover 4.11 0.36 3.43 4.86 1.00 6284
phi_NNWHUCover 3.06 0.26 2.59 3.60 1.00 5550
shape_Seedlings 0.29 0.04 0.23 0.36 1.00 3212
shape_Saplings 0.38 0.04 0.30 0.46 1.00 3476
Tail_ESS
phi_BGroundCover 3364
phi_NWUCover 2567
phi_NNWHUCover 3221
shape_Seedlings 2709
shape_Saplings 2621
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Deer Density Conditional Effects
Given our main interest in the model is deer density and how it impacts the suite of response variables, we generate marginal/conditional effects plots for the relationship between deer and the vegetation metric. Based on these plots we can see a “significant” (confidence intervals not-overlapping zero) effect of deer density on: bare ground (-ve), native woody understorey (-ve), native non-woody/herbaceous understorey (+ve), and presence of exotic species (+ve):
Show code
plot_list <- conditional_effects(vegfit, effects = "AllDeerDensity", prob = 0.9)
plot_list_50 <- conditional_effects(vegfit, effects = "AllDeerDensity", prob = 0.5)
plot_deer_relationship <- function(plot_data, plot_data_50, yaxis) {
ggplot(plot_data) +
geom_ribbon(aes(x = AllDeerDensity, ymin = `lower__`, ymax = `upper__`),
fill = delwp_cols[["Navy"]], alpha = 0.3) +
geom_ribbon(aes(x = AllDeerDensity, ymin = `lower__`, ymax = `upper__`),
fill = delwp_cols[["Navy"]], alpha = 0.3, data = plot_data_50) +
geom_smooth(aes(x = AllDeerDensity, y = `estimate__`),
colour = delwp_cols[["Navy"]], method = "loess", formula = y~x, linewidth = 2) +
ylab(paste(yaxis)) +
xlab("Deer Density (per km^2^)") +
delwp_theme() +
theme(axis.title.x = ggtext::element_markdown())
}
vegplot <- cowplot::plot_grid(plot_deer_relationship(plot_list[[1]], plot_list_50[[1]], "Bare Ground Cover"),
plot_deer_relationship(plot_list[[2]], plot_list_50[[2]], "Native Woody Understorey Cover"),
plot_deer_relationship(plot_list[[3]], plot_list_50[[3]], "Native Herbaceous Understorey Cover"),
plot_deer_relationship(plot_list[[4]], plot_list_50[[4]], "Seedling Counts") + scale_y_sqrt(),
plot_deer_relationship(plot_list[[5]], plot_list_50[[5]], "Sapling Counts"),
plot_deer_relationship(plot_list[[6]], plot_list_50[[6]], "Presence of Exotic Flora"),
ncol = 3, labels = "AUTO")
ggsave(plot = vegplot, filename = "outputs/plots/vegetation_effects_plot.png", width = 3400, height = 2200, units = "px")
vegplot
Figure 14: Impact of deer density on a range of vegetation measure, including (A) bare ground cover, (B) native woody understorey, (C) native herbaceous understorey cover, (D) seedlings, (E) saplings, and (F) th presence of exotic species. The line shows the mean estimate, with the shading representing 50% and 90% confidence intervals.
We found deer density to have several plausible relationships with several structural vegetation componentsw, including negative relationships with bare ground cover (\(\beta\) = -0.189 [90% CI: -0.337, -0.043]), native woody understorey cover (\(\beta\) = -0.181 [90% CI: -0.313, -0.047]), and a positive relationship with native non-woody/herbacous understorey cover (\(\beta\) = 0.21 [90% CI: 0.067, 0.355]). We also found strong evidence suggesting the presence of weeds is more likely at sites with higher densities of deer (\(\beta\) = 0.574 [90% CI: 0.232, 0.939], \(OR\) = 1.776 [90% CI: 1.261, 2.558]). We did not find tangible relationships between deer abundance and seedling or sapling counts.
Analysis Summary
Our studies detected deer at 148/317 cameras across Victoria and some form of deer sign (camera or transect) at 186/317 sites. Our average estimates of deer density at the 317 sites ranged from 0 to 28.3 deer per km2. Across approximately 74,570 km2 of public land in Victoria we estimate total deer abundance of the four species investigated in this study (Sambar, Fallow, Red and Hog) at 192,367 (90 % CI: 146,732 - 255,490. We also find plausible links between deer abundance and increased presence of exotic weed species.
Comparisons to other studies
Show code
asot_areas <- st_read("data/asot_operational_areas")Multiple layers are present in data source /Users/justincally/Documents/Work/R Repositories/statewide-deer-analysis/data/asot_operational_areas, reading layer `asot_operational_areas'.
Use `st_layers' to list all layer names and their type in a data source.
Set the `layer' argument in `st_read' to read a particular layer.
Reading layer `asot_operational_areas' from data source
`/Users/justincally/Documents/Work/R Repositories/statewide-deer-analysis/data/asot_operational_areas'
using driver `ESRI Shapefile'
Simple feature collection with 10 features and 1 field
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 2648000 ymin: 2401354 xmax: 2940500 ymax: 2609757
Projected CRS: GDA94 / VicgridShow code
asot_abundance <- exactextractr::exact_extract(combined_raster[[1:2]], y = asot_areas, "sum") %>%
`names<-`(c("Sambar_N", "Fallow_N"))
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